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1. The document provides revision on various topics in vectors including ratio theorem, scalar and vector products, lines, planes, perpendiculars, reflections, angles, distances, direction cosines, and geometric meanings. 2. Key concepts covered include using scalar and vector products to find angles between lines, planes, and determining if lines or planes are parallel/perpendicular. 3. Methods for finding the foot of a perpendicular from a point to a line or plane, reflecting lines and planes, and determining relationships between lines and planes are summarized.

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Vectors2

The document discusses various topics related to vectors and planes:
1. It explains the vector product in three forms - mathematical calculation, 3D picture, and in terms of sine. It provides an example to calculate the vector product of two vectors.
2. It discusses the different forms of the equation of a plane - parametric, scalar product, and Cartesian forms. It provides examples to write the equation of a plane in these different forms.
3. It explains how to find the foot of the perpendicular from a point to a plane. It provides examples to find the foot and shortest distance.
4. It discusses how to find acute angles between lines, planes, and a line and plane. Examples

Functions JC H2 Maths

This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.

JC H2 Physics Formula List/Summary (all topics)

Some of the concepts might be out of syllabus but I believe most of it is still relevant. It is a very concise summary containing mostly formulas and the Laws that govern Physics. This was done by a Raffles Institution student. I hope you will find this beneficial!

Ch 2 ~ vector

This document provides an overview of vectors and their applications in physics. It defines vectors and differentiates them from scalars, discusses vector notation and representation, and covers key concepts like addition, subtraction, and multiplication of vectors. Examples are given of vector quantities like displacement, velocity and force. The document also explains vector operators like gradient, divergence and curl, which allow converting between scalar and vector quantities, and outlines how calculus is important in physics for studying change.

Math for Physics Cheat sheet

This document provides a summary of basic math concepts for physics including:
1) Algebraic operations for solving equations for variables such as adding, subtracting, multiplying, and dividing terms.
2) Procedures for calculations with significant figures when adding, subtracting, multiplying, and dividing quantities.
3) How to write numbers in scientific notation and perform calculations using scientific notation such as multiplication, division, powers, and roots.

Lesson 7-8: Derivatives and Rates of Change, The Derivative as a function

The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative

Dimensions of Physical Quantities | Physics

The nature of physical quantity is described by nature of its dimensions. When we observe an object, the first thing we notice is the dimensions. In fact, we are also defined or observed with respect to our dimensions that is, height, weight, the amount of flesh etc. Copy the link given below and paste it in new browser window to get more information on Dimensions of Physical Quantities www.askiitians.com/iit-jee-physics/general-physics/dimensions-of-physical-quantities/

Torque

The document discusses torque and its relationship to force and moment arm. Torque is defined as the tendency to produce rotational motion and is calculated as the product of a force and its moment arm. Several examples are provided to illustrate calculating torque based on given forces and moment arms. The importance of moment arm in producing torque is that torque depends on both the magnitude of force and its distance from the axis of rotation.

Vectors2

The document discusses various topics related to vectors and planes:
1. It explains the vector product in three forms - mathematical calculation, 3D picture, and in terms of sine. It provides an example to calculate the vector product of two vectors.
2. It discusses the different forms of the equation of a plane - parametric, scalar product, and Cartesian forms. It provides examples to write the equation of a plane in these different forms.
3. It explains how to find the foot of the perpendicular from a point to a plane. It provides examples to find the foot and shortest distance.
4. It discusses how to find acute angles between lines, planes, and a line and plane. Examples

Functions JC H2 Maths

This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.

JC H2 Physics Formula List/Summary (all topics)

Some of the concepts might be out of syllabus but I believe most of it is still relevant. It is a very concise summary containing mostly formulas and the Laws that govern Physics. This was done by a Raffles Institution student. I hope you will find this beneficial!

Ch 2 ~ vector

This document provides an overview of vectors and their applications in physics. It defines vectors and differentiates them from scalars, discusses vector notation and representation, and covers key concepts like addition, subtraction, and multiplication of vectors. Examples are given of vector quantities like displacement, velocity and force. The document also explains vector operators like gradient, divergence and curl, which allow converting between scalar and vector quantities, and outlines how calculus is important in physics for studying change.

Math for Physics Cheat sheet

This document provides a summary of basic math concepts for physics including:
1) Algebraic operations for solving equations for variables such as adding, subtracting, multiplying, and dividing terms.
2) Procedures for calculations with significant figures when adding, subtracting, multiplying, and dividing quantities.
3) How to write numbers in scientific notation and perform calculations using scientific notation such as multiplication, division, powers, and roots.

Lesson 7-8: Derivatives and Rates of Change, The Derivative as a function

The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative

Dimensions of Physical Quantities | Physics

The nature of physical quantity is described by nature of its dimensions. When we observe an object, the first thing we notice is the dimensions. In fact, we are also defined or observed with respect to our dimensions that is, height, weight, the amount of flesh etc. Copy the link given below and paste it in new browser window to get more information on Dimensions of Physical Quantities www.askiitians.com/iit-jee-physics/general-physics/dimensions-of-physical-quantities/

Torque

The document discusses torque and its relationship to force and moment arm. Torque is defined as the tendency to produce rotational motion and is calculated as the product of a force and its moment arm. Several examples are provided to illustrate calculating torque based on given forces and moment arms. The importance of moment arm in producing torque is that torque depends on both the magnitude of force and its distance from the axis of rotation.

Vectors

What are vectors? How to add and subtract vectors using graphics and components.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f

Movimiento en dos y tres dimensiones

This document discusses vectors and vector addition in two and three dimensions. It provides examples of displacement vectors, distance traveled, and the relationship between the two. It also contains problems calculating vector components, magnitudes, and directions in various scenarios involving particle motion along paths and circles. Solutions are provided for each multi-part problem.

Physics a2 unit4_06_centripetal_force -centripetal force

This document discusses centripetal acceleration and centripetal force. It defines centripetal acceleration as acceleration toward the center of a circular path caused by changing velocity. An equation is given for centripetal acceleration using angular velocity and radius. It also defines centripetal force as the force causing an object to travel in a circular path, and gives an equation for centripetal force using mass, velocity, and radius. Examples are provided to demonstrate calculating speed, acceleration, and force for objects moving in circular motion.

2. Vector Calculus.ppt

The document discusses vector calculus concepts including:
- Vector product has magnitude equal to the area of the parallelogram formed by the vectors and is perpendicular to both vectors.
- Scalar triple product equals the volume of the parallelepiped formed by the three vectors and represents the scalar quantity obtained by multiplying the vectors in a specific order.
- Vector triple product follows the "bac-cab" rule and results in another vector quantity.

Motion, Scalar and Vector

This document defines and discusses key concepts related to motion, including reference points, distance, displacement, scalar and vector quantities. It notes that motion depends on the choice of reference point. Distance is the length of the path traveled between two positions, while displacement is a vector quantity referring to the straight line distance between two positions. Scalar quantities only have magnitude, while vector quantities have both magnitude and direction. Examples of each are provided.

Gravitational field and potential, escape velocity, universal gravitational l...

What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples

The washer method

The document discusses methods for calculating the volume of solids of revolution. It reviews the disk method for revolving regions about a horizontal axis and introduces the washer method for regions with holes. Examples are provided for using these methods to set up integrals and calculate volumes when revolving about horizontal and vertical axes. The document also discusses how these techniques can be generalized to solids with non-circular cross-sections of known area formulas.

Scalar and vector quantities

This document discusses physical quantities and vectors. It defines two types of physical quantities: scalar quantities which have only magnitude, and vector quantities which have both magnitude and direction. Examples of each are given. Vector quantities are represented by magnitude and direction. The document then discusses methods for adding and subtracting vectors graphically using head-to-tail and parallelogram methods. It also covers resolving vectors into rectangular components, finding the magnitude and direction of vectors, dot products of vectors which yield scalar quantities, and cross products of vectors which yield vector quantities. Examples of applying these vector concepts are provided.

Newton’s law of gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula for this gravitational force is F = G(m1m2)/r^2, where F is the force of attraction, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass. The document provides examples of applying this formula to calculate gravitational forces between people, people and Earth, and people and objects.

Vector&scalar quantitiesppt

The document discusses the key differences between scalar and vector quantities, explaining that scalars only have magnitude while vectors have both magnitude and direction, and provides examples of each. It then focuses on methods for representing and adding vectors graphically and mathematically, including tip-to-tail addition, component methods, and subtraction by adding the opposite vector.

Ch 12 Temperature and Heat

The document discusses key concepts in temperature and heat, including:
1. It introduces common temperature scales like Fahrenheit, Celsius, and Kelvin scales. It explains how each scale was developed and their distinguishing features.
2. It discusses concepts like thermal expansion - both linear and volumetric expansion. Linear expansion explains how the length of an object changes with temperature, while volumetric expansion explains how the volume changes.
3. It provides examples of calculating temperature conversions between different scales, as well as examples of using equations of linear and volumetric expansion to solve problems involving changes in length or volume due to temperature changes.

Work And Energy

It takes energy to do work on an object by exerting a force over a distance. Work is measured in joules, which is the energy required to lift a quarter pound cheeseburger from a table to above one's head. Power is the rate at which work is done and is measured in watts. There are different forms of mechanical energy including potential energy stored by an object's position or shape, and kinetic energy based on an object's motion. The total amount of energy in the universe remains constant according to the law of conservation of energy, even as it transforms between different forms.

Rotational motion (3)

This document defines and explains various concepts related to rotational motion including:
1. Angular position, displacement, velocity, and acceleration which describe the rotational analogs of linear position, displacement, velocity, and acceleration.
2. Key equations of angular motion analogous to linear equations of motion.
3. Relationships between linear and angular quantities like displacement, velocity, and acceleration.
4. Concepts like moment of inertia, radius of gyration, angular momentum, and torque which describe rotational motion and its relationship to applied forces.
5. Theorems regarding moment of inertia including the theorem of parallel axes and perpendicular axes.

Relative velocity

The document provides an overview of the objectives and activities for a lesson on relative motion analysis. It includes sample problems and questions on determining relative position, velocity, and acceleration between two moving frames of reference using vector methods and trigonometric relationships like the laws of sines and cosines. Sample problems demonstrate how to set up and solve for unknown relative motion variables graphically or through vector equations.

Ejercicio de cinética de un punto material (Quiz-Mayo 2017)

La bomba centrífuga gira a una velocidad angular ω. Se calcula la fuerza N ejercida por uno de los álabes sobre una partícula de masa m cuando se mueve hacia afuera a lo largo del álabe desde el centro R=0 hasta un radio R. La solución muestra que la trayectoria de la partícula es una hipérbola y usa las ecuaciones de aceleración radial y angular para derivar que la fuerza N es igual a mω2R.

maths.ppt

The document discusses several real-world applications of differential and integral calculus. It provides examples of first-order differential equations being used to model jumping motions in video games and the cooling of objects. Surface and volume integrals are applied in fields like electrostatics, fluid dynamics, and continuity equations. Matrix determinants can estimate areas like that of the Bermuda Triangle. Overall, calculus has wide applications in science, engineering, economics and other domains.

Work force energy ppt final wiki

The document discusses various types of energy and forces, explaining that energy cannot be created or destroyed, only changed in form, and defines work as the ability to cause change when a force acts over a distance. It also explains Newton's three laws of motion, including that an object at rest or in motion will remain as such unless acted on by an unbalanced force, and that for every action there is an equal and opposite reaction. The document provides examples and formulas to help understand these concepts of energy, forces, and motion.

Vectors and Kinematics

Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1

Momentum and Energy.pptx

1) The document discusses concepts related to mechanics including momentum, energy, and collisions. It provides examples and questions to illustrate these concepts.
2) Key ideas covered include the definitions of momentum and velocity, the conservation of momentum especially during collisions, different types of energy and the law of conservation of energy, and how energy and momentum can be used to analyze motion and collisions.
3) Examples include analyzing the final velocity of two cars that collide and stick together, calculating the velocity of an egg after being dropped from a building using conservation of energy, and determining the velocities after both perfectly elastic and inelastic collisions.

PHY300 Chapter 5 physics 5e

This chapter of the physics textbook discusses circular motion. It introduces concepts like angular displacement, angular velocity, angular acceleration, radian measure, and their relationships to linear displacement, velocity, and acceleration. It describes uniform circular motion and the radial (centripetal) acceleration required. Examples are provided to demonstrate calculating angular speed, period, frequency, and the force required for uniform circular motion. Rolling motion and projectile motion on a circular path are also discussed.

Three dimensional geometry

This document defines direction cosines and ratios of a line, and discusses how to find them given information about the line. It also defines planes and their equations in different forms, including the normal form using distance from origin and direction cosines of the normal vector, and the form passing through a point perpendicular to a given direction. It further discusses finding the angle between lines or between a line and plane.

How to design a linear control system

How to design a linear control system?
in this article you can learn designing of a linear control system.

Vectors

What are vectors? How to add and subtract vectors using graphics and components.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f

Movimiento en dos y tres dimensiones

This document discusses vectors and vector addition in two and three dimensions. It provides examples of displacement vectors, distance traveled, and the relationship between the two. It also contains problems calculating vector components, magnitudes, and directions in various scenarios involving particle motion along paths and circles. Solutions are provided for each multi-part problem.

Physics a2 unit4_06_centripetal_force -centripetal force

This document discusses centripetal acceleration and centripetal force. It defines centripetal acceleration as acceleration toward the center of a circular path caused by changing velocity. An equation is given for centripetal acceleration using angular velocity and radius. It also defines centripetal force as the force causing an object to travel in a circular path, and gives an equation for centripetal force using mass, velocity, and radius. Examples are provided to demonstrate calculating speed, acceleration, and force for objects moving in circular motion.

2. Vector Calculus.ppt

The document discusses vector calculus concepts including:
- Vector product has magnitude equal to the area of the parallelogram formed by the vectors and is perpendicular to both vectors.
- Scalar triple product equals the volume of the parallelepiped formed by the three vectors and represents the scalar quantity obtained by multiplying the vectors in a specific order.
- Vector triple product follows the "bac-cab" rule and results in another vector quantity.

Motion, Scalar and Vector

This document defines and discusses key concepts related to motion, including reference points, distance, displacement, scalar and vector quantities. It notes that motion depends on the choice of reference point. Distance is the length of the path traveled between two positions, while displacement is a vector quantity referring to the straight line distance between two positions. Scalar quantities only have magnitude, while vector quantities have both magnitude and direction. Examples of each are provided.

Gravitational field and potential, escape velocity, universal gravitational l...

What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples

The washer method

The document discusses methods for calculating the volume of solids of revolution. It reviews the disk method for revolving regions about a horizontal axis and introduces the washer method for regions with holes. Examples are provided for using these methods to set up integrals and calculate volumes when revolving about horizontal and vertical axes. The document also discusses how these techniques can be generalized to solids with non-circular cross-sections of known area formulas.

Scalar and vector quantities

This document discusses physical quantities and vectors. It defines two types of physical quantities: scalar quantities which have only magnitude, and vector quantities which have both magnitude and direction. Examples of each are given. Vector quantities are represented by magnitude and direction. The document then discusses methods for adding and subtracting vectors graphically using head-to-tail and parallelogram methods. It also covers resolving vectors into rectangular components, finding the magnitude and direction of vectors, dot products of vectors which yield scalar quantities, and cross products of vectors which yield vector quantities. Examples of applying these vector concepts are provided.

Newton’s law of gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula for this gravitational force is F = G(m1m2)/r^2, where F is the force of attraction, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass. The document provides examples of applying this formula to calculate gravitational forces between people, people and Earth, and people and objects.

Vector&scalar quantitiesppt

The document discusses the key differences between scalar and vector quantities, explaining that scalars only have magnitude while vectors have both magnitude and direction, and provides examples of each. It then focuses on methods for representing and adding vectors graphically and mathematically, including tip-to-tail addition, component methods, and subtraction by adding the opposite vector.

Ch 12 Temperature and Heat

The document discusses key concepts in temperature and heat, including:
1. It introduces common temperature scales like Fahrenheit, Celsius, and Kelvin scales. It explains how each scale was developed and their distinguishing features.
2. It discusses concepts like thermal expansion - both linear and volumetric expansion. Linear expansion explains how the length of an object changes with temperature, while volumetric expansion explains how the volume changes.
3. It provides examples of calculating temperature conversions between different scales, as well as examples of using equations of linear and volumetric expansion to solve problems involving changes in length or volume due to temperature changes.

Work And Energy

It takes energy to do work on an object by exerting a force over a distance. Work is measured in joules, which is the energy required to lift a quarter pound cheeseburger from a table to above one's head. Power is the rate at which work is done and is measured in watts. There are different forms of mechanical energy including potential energy stored by an object's position or shape, and kinetic energy based on an object's motion. The total amount of energy in the universe remains constant according to the law of conservation of energy, even as it transforms between different forms.

Rotational motion (3)

This document defines and explains various concepts related to rotational motion including:
1. Angular position, displacement, velocity, and acceleration which describe the rotational analogs of linear position, displacement, velocity, and acceleration.
2. Key equations of angular motion analogous to linear equations of motion.
3. Relationships between linear and angular quantities like displacement, velocity, and acceleration.
4. Concepts like moment of inertia, radius of gyration, angular momentum, and torque which describe rotational motion and its relationship to applied forces.
5. Theorems regarding moment of inertia including the theorem of parallel axes and perpendicular axes.

Relative velocity

The document provides an overview of the objectives and activities for a lesson on relative motion analysis. It includes sample problems and questions on determining relative position, velocity, and acceleration between two moving frames of reference using vector methods and trigonometric relationships like the laws of sines and cosines. Sample problems demonstrate how to set up and solve for unknown relative motion variables graphically or through vector equations.

Ejercicio de cinética de un punto material (Quiz-Mayo 2017)

La bomba centrífuga gira a una velocidad angular ω. Se calcula la fuerza N ejercida por uno de los álabes sobre una partícula de masa m cuando se mueve hacia afuera a lo largo del álabe desde el centro R=0 hasta un radio R. La solución muestra que la trayectoria de la partícula es una hipérbola y usa las ecuaciones de aceleración radial y angular para derivar que la fuerza N es igual a mω2R.

maths.ppt

The document discusses several real-world applications of differential and integral calculus. It provides examples of first-order differential equations being used to model jumping motions in video games and the cooling of objects. Surface and volume integrals are applied in fields like electrostatics, fluid dynamics, and continuity equations. Matrix determinants can estimate areas like that of the Bermuda Triangle. Overall, calculus has wide applications in science, engineering, economics and other domains.

Work force energy ppt final wiki

The document discusses various types of energy and forces, explaining that energy cannot be created or destroyed, only changed in form, and defines work as the ability to cause change when a force acts over a distance. It also explains Newton's three laws of motion, including that an object at rest or in motion will remain as such unless acted on by an unbalanced force, and that for every action there is an equal and opposite reaction. The document provides examples and formulas to help understand these concepts of energy, forces, and motion.

Vectors and Kinematics

Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1

Momentum and Energy.pptx

1) The document discusses concepts related to mechanics including momentum, energy, and collisions. It provides examples and questions to illustrate these concepts.
2) Key ideas covered include the definitions of momentum and velocity, the conservation of momentum especially during collisions, different types of energy and the law of conservation of energy, and how energy and momentum can be used to analyze motion and collisions.
3) Examples include analyzing the final velocity of two cars that collide and stick together, calculating the velocity of an egg after being dropped from a building using conservation of energy, and determining the velocities after both perfectly elastic and inelastic collisions.

PHY300 Chapter 5 physics 5e

This chapter of the physics textbook discusses circular motion. It introduces concepts like angular displacement, angular velocity, angular acceleration, radian measure, and their relationships to linear displacement, velocity, and acceleration. It describes uniform circular motion and the radial (centripetal) acceleration required. Examples are provided to demonstrate calculating angular speed, period, frequency, and the force required for uniform circular motion. Rolling motion and projectile motion on a circular path are also discussed.

Vectors

Vectors

Movimiento en dos y tres dimensiones

Movimiento en dos y tres dimensiones

Physics a2 unit4_06_centripetal_force -centripetal force

Physics a2 unit4_06_centripetal_force -centripetal force

2. Vector Calculus.ppt

2. Vector Calculus.ppt

Motion, Scalar and Vector

Motion, Scalar and Vector

Gravitational field and potential, escape velocity, universal gravitational l...

Gravitational field and potential, escape velocity, universal gravitational l...

The washer method

The washer method

Scalar and vector quantities

Scalar and vector quantities

Newton’s law of gravitation

Newton’s law of gravitation

Vector&scalar quantitiesppt

Vector&scalar quantitiesppt

Ch 12 Temperature and Heat

Ch 12 Temperature and Heat

Work And Energy

Work And Energy

Rotational motion (3)

Rotational motion (3)

Relative velocity

Relative velocity

Ejercicio de cinética de un punto material (Quiz-Mayo 2017)

Ejercicio de cinética de un punto material (Quiz-Mayo 2017)

maths.ppt

maths.ppt

Work force energy ppt final wiki

Work force energy ppt final wiki

Vectors and Kinematics

Vectors and Kinematics

Momentum and Energy.pptx

Momentum and Energy.pptx

PHY300 Chapter 5 physics 5e

PHY300 Chapter 5 physics 5e

Three dimensional geometry

This document defines direction cosines and ratios of a line, and discusses how to find them given information about the line. It also defines planes and their equations in different forms, including the normal form using distance from origin and direction cosines of the normal vector, and the form passing through a point perpendicular to a given direction. It further discusses finding the angle between lines or between a line and plane.

How to design a linear control system

How to design a linear control system?
in this article you can learn designing of a linear control system.

Three dim. geometry

Three key points about three-dimensional geometry from the document:
1) Three-dimensional geometry developed in accordance with Einstein's field equations and is useful in fields like electromagnetism and for constructing 3D models using computer algorithms.
2) The document presents a vector-algebra approach to three-dimensional geometry, defining points as ordered triples of real numbers and discussing properties of lines and planes.
3) Key concepts discussed include the vector and Cartesian equations of lines and planes, direction cosines and ratios, angles between lines, perpendicularity, parallelism, and intersections. Formulas are provided for distances, divisions, and reflections.

Solution kepler chap 1

This document is a solution to a physics problem set composed and formatted by E.A. Baltz and M. Strovink. It contains solutions to 6 problems using vector algebra and trigonometry. The document uses concepts like the law of cosines, dot products, cross products, and vector identities to break vectors into components and calculate angles between vectors. It also applies these concepts to problems involving vectors representing locations on a sphere and wind resistance problems for airplanes.

Class 14 3D HermiteInterpolation.pptx

1) Interpolation is a process that defines a function to connect data points by determining a unique polynomial curve that passes through the specified points.
2) Cubic spline interpolation fits piecewise cubic polynomials to pass through given control points, with the curves matching positions and derivatives at the points.
3) Hermite interpolation also uses piecewise cubics but specifies the tangent at each control point, allowing local control of the curve sections. Boundary conditions define the curves to match positions and specified tangent derivatives at endpoints.

Notes on Equation of Plane

This document provides notes on determining various properties of planes in 3D space, including:
1) The perpendicular distance from a point to a plane using either vector or Cartesian methods.
2) The angle between a plane and line by taking the arcsine of the dot product of their normal vectors.
3) The angle between two planes by taking the arccosine of the dot product of their normal vectors.
Worked examples are provided for calculating distances, angles, and deriving relevant formulas. Revision questions at the end reinforce the content through calculation practice.

5.vector geometry Further Mathematics Zimbabwe Zimsec Cambridge

vector geometry Further Mathematics Zimbabwe Zimsec Cambridge
Zimsec
Zimbabwe
Alpro Elearning Portal

Higher Maths 1.1 - Straight Line

The document discusses various concepts relating to straight lines in mathematics including:
1) Calculating the gradient of a straight line between two points.
2) Horizontal and vertical lines having gradients of 0 or being undefined.
3) The relationship between gradient and angle of a line.
4) Finding the midpoint, collinearity of points, and gradients of perpendicular lines.

Math - analytic geometry

1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.

Vectors.pdf

The document defines a vector as having both magnitude and direction, represented geometrically by an arrow. It discusses representing vectors algebraically using coordinates, and defines operations like addition, subtraction, and scaling of vectors. Key vector concepts covered include the dot product, which yields a scalar when combining two vectors, and unit vectors, which have a magnitude of 1. Examples are provided of using vectors to solve problems and prove geometric properties.

Vectors.pdf

The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar value that geometrically equals the magnitudes of A and B multiplied by the cosine of the angle between them.

Vectors.pdf

The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar equal to |A||B|cosθ, where θ is the angle between the vectors.

Determinants

This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.

7.5 lines and_planes_in_space

Here are the steps to find the line of intersection of the two planes:
1) Write the equations of the planes in standard form:
Plane 1: x + 2y - z = 4
Plane 2: 2x - y + z = 1
2) Set the equations equal to each other and solve as a system of equations:
x + 2y - z = 4
2x - y + z = 1
3) Eliminate one variable:
Subtract the second equation from the first:
(x + 2y - z) - (2x - y + z) = 4 - 1
-x + y = 3
4) Substitute back into one of the

Relative squared distances to a conic

The midpoint method or technique is a “measurement” and as each measurement it has a tolerance, but
worst of all it can be invalid, called Out-of-Control or OoC. The core of all midpoint methods is the accurate
measurement of the difference of the squared distances of two points to the “polar” of their midpoint
with respect to the conic. When this measurement is valid, it also measures the difference of the squared
distances of these points to the conic, although it may be inaccurate, called Out-of-Accuracy or OoA. The
primary condition is the necessary and sufficient condition that a measurement is valid. It is comletely
new and it can be checked ultra fast and before the actual measurement starts. .
Modeling an incremental algorithm, shows that the curve must be subdivided into “piecewise monotonic”
sections, the start point must be optimal, and it explains that the 2D-incremental method can find, locally,
the global Least Square Distance. Locally means that there are at most three candidate points for a given
monotonic direction; therefore the 2D-midpoint method has, locally, at most three measurements.
When all the possible measurements are invalid, the midpoint method cannot be applied, and in that case
the ultra fast “OoC-rule” selects the candidate point. This guarantees, for the first time, a 100% stable,
ultra-fast, berserkless midpoint algorithm, which can be easily transformed to hardware. The new algorithm
is on average (26.5±5)% faster than Mathematica, using the same resolution and tested using 42
different conics. Both programs are completely written in Mathematica and only ContourPlot[] has been
replaced with a module to generate the grid-points, drawn with Mathematica’s
Graphics[Line{gridpoints}] function.

Lab mannual ncert 3

1. A model is constructed using plywood, nails, and wires to represent two skew lines in three-dimensional space.
2. The shortest distance between the two skew lines is measured directly using a ruler.
3. The coordinates of points on the two lines are used to calculate the shortest distance analytically via a formula.
4. The distances obtained through measurement and calculation are found to be approximately the same, verifying the analytical method.

Calculus academic journal (sample)

1. The document discusses techniques for finding extrema of functions, including absolute and local extrema. Critical points, endpoints, and the first and second derivative tests are covered.
2. The mean value theorem and Rolle's theorem are summarized. The mean value theorem relates the average and instantaneous rates of change over an interval.
3. Optimization problems can be solved by setting the derivative of the objective function equal to zero to find critical points corresponding to maxima or minima.
4. Newton's method is presented as an iterative process for approximating solutions to equations, using tangent lines to generate a sequence of improving approximations.
5. Anti-derivatives are defined as functions whose derivatives are a given

Analytical geometry slides

The document discusses equations of lines, including:
1) The gradient-point form of a straight line equation which uses the gradient and coordinates of one point to determine the equation.
2) Calculating the gradient from two points on a line and using it to find the angle of inclination.
3) Determining the equation of a line parallel to another line, by setting their gradients equal since parallel lines have the same gradient.

Additional Mathematics form 4 (formula)

This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.

Spm add-maths-formula-list-form4-091022090639-phpapp01

This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.

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How to design a linear control system

How to design a linear control system

Three dim. geometry

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Determinants

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Analytical geometry slides

Additional Mathematics form 4 (formula)

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Spm add-maths-formula-list-form4-091022090639-phpapp01

Spm add-maths-formula-list-form4-091022090639-phpapp01

Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Sec 4 A Maths Notes Maxima Minima

The document contains two examples of maximum and minimum problems involving differentiation.
Example 1 asks the reader to find the minimum volume of a cone given that a sphere must fit inside it. It is found that the minimum volume occurs when the radius of the cone is 28.577 cm.
Example 2 involves finding the maximum volume of a cylinder inscribed in a sphere. The maximum volume is calculated to be 104,000 cm3, occurring when the height of the cylinder is 28.5 cm.
The document provides guidance on solving maximum and minimum problems using differentiation, illustrated through these two examples involving geometric shapes.

Sec 3 E Maths Notes Coordinate Geometry

This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

Sec 2 Maths Notes Change Subject

The document provides examples and techniques for changing the subject of a formula. It demonstrates flipping both sides of an equation, multiplying or dividing both sides by the same term, and isolating the term to be made the subject by collecting like terms on one side of the equation and leaving the term by itself on the other side. Common mistakes discussed include incorrectly flipping terms individually rather than both sides of the equation and prematurely making denominators the same.

Sec 1 Maths Notes Equations

1) The document provides steps for solving simple linear equations with no fractions and fractional equations.
2) For linear equations with no fractions, the steps are to expand if there is a bracket, group like terms to one side, and then solve for the variable.
3) For fractional equations, the steps are to multiply both sides by the common denominator to clear the fractions, then group like terms and solve.

Math academy-partial-fractions-notes

This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.

Complex Numbers 1 - Math Academy - JC H2 maths A levels

The document provides lessons on complex numbers. It defines a complex number as being of the form z = x + iy, where x and y are real numbers. It discusses operations like addition, subtraction, multiplication and division of complex numbers. It also defines the complex conjugate and gives some examples of performing operations on complex numbers.

Recurrence

This document provides examples of recurrence relations and their solutions. It begins by defining convergence of sequences and limits. It then provides examples of recurrence relations, solving them using algebraic and graphical methods. One example finds the 6th term of a sequence defined by a recurrence relation to be 2.3009. Another example solves a recurrence relation algebraically to express the general term un in terms of n. The document emphasizes using graphical methods like sketching graphs to prove properties of sequences defined by recurrence relations.

Functions 1 - Math Academy - JC H2 maths A levels

Slides for Functions lecture 1.
JC H2 Maths
A levels Singapore
www.mathacademy.sg
Copyright 2015 Math Academy

Probability 2 - Math Academy - JC H2 maths A levels

The document provides information on probability concepts including Venn diagrams, union and intersection of events, useful probability formulas, mutually exclusive vs independent events, and examples testing these concepts. Specifically, it defines union as "taking everything in A and B", intersection as "taking common parts in A and B", provides formulas for probability of unions and intersections, and shows how to determine if events are mutually exclusive or independent using the probability of their intersection. It also includes worked examples testing concepts like mutually exclusive events and independence.

Probability 1 - Math Academy - JC H2 maths A levels

The document discusses conditional probability and provides examples. It defines conditional probability P(A|B) as the probability of event A occurring given that event B has already occurred. An example calculates probabilities for drawing marbles from a bag. Another example finds probabilities for selecting chocolates with different flavors from a box containing chocolates of various flavors. Formulas and step-by-step workings are provided for calculating conditional probabilities.

Sec 3 A Maths Notes Indices

Sec 3 A Maths Notes Indices

Sec 4 A Maths Notes Maxima Minima

Sec 4 A Maths Notes Maxima Minima

Sec 3 E Maths Notes Coordinate Geometry

Sec 3 E Maths Notes Coordinate Geometry

Sec 3 A Maths Notes Indices

Sec 3 A Maths Notes Indices

Sec 2 Maths Notes Change Subject

Sec 2 Maths Notes Change Subject

Sec 1 Maths Notes Equations

Sec 1 Maths Notes Equations

Math academy-partial-fractions-notes

Math academy-partial-fractions-notes

Complex Numbers 1 - Math Academy - JC H2 maths A levels

Complex Numbers 1 - Math Academy - JC H2 maths A levels

Recurrence

Recurrence

Functions 1 - Math Academy - JC H2 maths A levels

Functions 1 - Math Academy - JC H2 maths A levels

Probability 2 - Math Academy - JC H2 maths A levels

Probability 2 - Math Academy - JC H2 maths A levels

Probability 1 - Math Academy - JC H2 maths A levels

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Stack Memory Organization of 8086 Microprocessor

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THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...

The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.

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Nutrition Inc FY 2024, 4 - Hour Training

Slides for Lessons: Homes and Centers

Chapter wise All Notes of First year Basic Civil Engineering.pptx

Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1

skeleton System.pdf (skeleton system wow)

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ملزمة تشريح الجهاز الهيكلي (نظري 3)
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تتميز هذهِ الملزمة بعِدة مُميزات :
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2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
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Electric Fetus - Record Store Scavenger Hunt

Electric Fetus is a record store in Minneapolis, MN

Temple of Asclepius in Thrace. Excavation results

The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).

spot a liar (Haiqa 146).pptx Technical writhing and presentation skills

sample presentation

Skimbleshanks-The-Railway-Cat by T S Eliot

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This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf

مصحف أحرف الخلاف للقراء العشرةأعد أحرف الخلاف بالتلوين وصلا سمير بسيوني غفر الله له

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Plenary presentation at the NTTC Inter-university Workshop, 18 June 2024, Manila Prince Hotel.

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Wound healing PPT

This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.

NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx

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Contains guides on answering questions on the specimens provided

Stack Memory Organization of 8086 Microprocessor

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Nutrition Inc FY 2024, 4 - Hour Training

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spot a liar (Haiqa 146).pptx Technical writhing and presentation skills

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- 1. Math Academy: Vectors Revision 1 Ratio Theorem µ λA P B O If −→ AP : −−→ PB = µ : λ, then −−→ OP = µ −−→ OB + λ −→ OA λ + µ . 2 Scalar Product a b θ a · b = |a||b| cos θ. Note that the direction of the 2 vectors must be as shown above. Properties of scalar product Note the difference between the first 2 points, with vector product. (a) (i) a · a = |a|2 (ii) a · b = b · a (iii) a · (b + c) = a · b + a · c (iv) a · λb = λ(a · b) = (λa) · b (b) Perpendicular Vectors If a and b are perpendicular, then a · b = 0. (c) Length of projection θ a b |a · ˆb| (d) Projection vector Projection vector of a onto b is given by (a · ˆb)ˆb or ( a · b |b| ) b |b| . 3 Vector Product a × b = (|a||b| sin θ)ˆn |a × b| = |a||b| sin θ Properties of vector product Note the difference between the first 2 points, with scalar product. (a) (i) a × a = 0 (ii) a × b = −b × a (iii) a × (b + c) = a × b + a × c (iv) a × λb = λ(a × b) = (λa) × b (b) Area of Triangle A B C h θ 1 2 −−→ AB × −−→ BC = 1 2 |cross product of two adjacent sides| (c) Area of Parallelogram A B C h θ D −−→ AB × −−→ BC = |cross product of two adjacent sides| 4 Lines Vector Equation Parametric: r = a + λm, λ ∈ R Cartesian: x − a1 m1 = y − a2 m2 = z − a3 m3 . Note: Ensure that you know how to change from vector to cartesian and vice versa. www.MathAcademy.sg 1 © 2019 Math Academy
- 2. Math Academy: Vectors Revision Relationship between two lines l1 : r = a1 + λm1 l2 : r = a2 + µm2 a) Case 1: Parallel lines Step 1: To check parallel: Check if m1 is a scalar multiple of m2. Step 2: If parallel, check if they are the same line: Take a point from ℓ1 and see if it lies on ℓ2. (i) If the point is in ℓ2, they are the same line. (ii) Otherwise, they are parallel, non- intersecting lines. Make sure you know how to show the working! (re- fer to main notes) b) Case 2: Intersecting/Skew lines Method 1: Equate both equations together, use GC to solve. No solution =⇒ Skew lines Solution found for λ and µ =⇒ Intersecting lines Method 2: Step 1: Equate both equations together, solve for the i and j - components. Step 2: Sub into k- component. See if it satisfies this component. Satisfies =⇒ intersecting. No satisfy =⇒ skew. This is the same way to find point of intersection of 2 lines. 5 Planes Parametric: r = −→ OA + λm1 + µm2 Scalar-Product: r · n = a · n Cartesian: ax + by + cz = D Ensure you know how to switch from parametric to scalar-prod to cartesian and cartesian to scalar-prod. You DONT have to know how to switch from scalar- prod/cartesian to parametric. 6 Foot of Perpendicular a) From point to line Given: (1) −−→ OP (2) ℓ : r = −→ OA + λm, λ ∈ R. We find the foot from point P to line ℓ. Since −−→ OF lies on the line ℓ, −−→ OF = −→ OA + λm, for some λ ∈ R A P F ℓ m O −−→ PF · m = 0 ( −−→ OF − −−→ OP) · m = 0 ( −→ OA + λm − −−→ OP) · m = 0. We solve the only unknown, λ, and substitute back into the equation −−→ OF = −→ OA + λm. b) From point to plane P N n Π Given point P and equation of the plane Π : r · n = D (1) Step 1: Form the equation of the line ℓ that passes through P and is perpendicular to Π. ℓ : r = −−→ OP + λn, λ ∈ R. (2) Step 2: Intersect ℓ with Π to get the foot of the perpendicular. That is, substitute (2) into (1). ( −−→ OP + λn) · n = D Substitute λ back into (2) to get the foot of the perpendicular. www.MathAcademy.sg 2 © 2019 Math Academy
- 3. Math Academy: Vectors Revision 7 Reflections In general, we need to first find reflection of a point. a) Reflect a point in a line/plane. We make use of foot of perpendicular and mid point theorem. A P P′ F ℓ O −−→ OF = −−→ OP + −−→ OP′ 2 . Make −−→ OP′ the subject. b) Reflect l1 in the line l2. ℓ2 P F P′ ℓ1 A (i) Form the new direction vector −−→ AP′ . (ii) Form the new equation using r = −→ OA + λ −−→ AP′ The same technique holds for reflection of line in a plane. c) Reflection of a plane in another plane. π1 π2 l Suppose we want to find the reflection of plane π1 in π2. Lets call it π′ 1. We also know that π1 and π2 intersect at the line l. 1) l will also lie on the reflected plane π′ 1. Hence π′ 1 also contains direction vector of l. 2) Take a point P from π1 and reflect it in π2. This reflected point P′ will be on π′ 1. Now take a point A from l (which is also on π′ 1). π′ 1 will contain direction vector −−→ AP′ . www.MathAcademy.sg 3 © 2019 Math Academy
- 4. Math Academy: Vectors Revision 8 Angles Note: For all of the following, if the question asks for ACUTE angle, you need to put modulus at the RHS, that is, at the dot product. ϕ θ Π ℓ m n Acute angle between line and a plane sin θ = m · n |m||n| . Π1 Π2 θ θ n2 n1 Acute angle between 2 planes cos θ = n1 · n2 |n1||n2| . θ ℓ1 ℓ2 m1 m2 Acute angle between two lines cos θ = m1 · m2 |m1||m2| . 9 Distance involving lines ℓ : r = a + λmA B θ | −−→ AB × ˆm| | −−→ AB · ˆm| 10 Distance involving planes Distance between point and plane B A n Π F θ | −−→ AB · ˆn| | −−→ AB × ˆn| Distance between parallel line/plane with plane A B | −−→ AB · ˆn| Π F ℓ A B | −−→ AB · ˆn| Π2 F Π1 | −−→ AB × ˆn| | −−→ AB × ˆn| www.MathAcademy.sg 4 © 2019 Math Academy
- 5. Math Academy: Vectors Revision 11 Distances from Origin Distance from origin to plane If r · ˆn = d, then, Distance from origin to plane = |d| Ensure that the equation is ˆn, not n! Π1 Π2 O d1 d2 n Distance between 2 parallel planes Π1 : r · ˆn = d1 Π2 : r · ˆn = d2 Distance between the two planes = |d1 − d2|. Note: If d1 and d2 are of opposite signs, then they lie on opposite sides of the origin. 12 Relationship between line and plane n ℓ : r = a + λm m Π : r · n = d (a) If a line and plane are parallel, m · n = 0 To check further if the line is ON the plane, we sub the equation of the line into the plane, see if it satisfies the equation. (refer to notes) (b) If a line and plane are perpendicular, m is parallel to n =⇒ m = kn for some constant k ∈ R. 13 Relationship between two planes (a) Parallel planes Two planes are parallel to each other ⇐⇒ Their normals are scalar multiple of each other. (b) Non-Parallel planes Any 2 non parallel, non identical planes will inter- sect in a line. n1 n2 ℓ Π1 Π2 The direction vector, m, of the line of intersection between Π1 and Π2 is given by m = n1 × n2, where n1, n2 are the normal vectors of Π1 and Π2 respectively. However! We will use a GC if there are no unknowns in n1 and n2. (c) Two perpendicular planes p1 p2 n2n1 If 2 planes, p1 and p2 are perpendicular, then the following occurs: (a) n1 is parallel to p2 =⇒ p2 contains the direc- tion vector n1, (b) n2 is parallel to p1 =⇒ p1 contains the direction vector n2. www.MathAcademy.sg 5 © 2019 Math Academy
- 6. Math Academy: Vectors Revision 14 Directional Cosines x y z α β γ Let the angles made with the x, y, and z-axes be θ, ϕ, and ω respectively. x-axis: cos θ = α √ α2 + β2 + γ2 y-axis: cos ϕ = β √ α2 + β2 + γ2 z-axis: cos ω = γ √ α2 + β2 + γ2 15 On Geometrical Meanings Recall the following diagram on projections: b a |a · ˆb| |a × ˆb| θ Lets now deal with geometrical interpretations. (a) |a · ˆb| (b) |a × ˆb| (c) (a · ˆb)ˆb Let b be any non-zero vector and c a unit vector, give the geometrical meaning of |c · b|. Let a, d be any 2 non-zero vectors. (d) Give a geometrical meaning of |a × d|. (e) Suppose we have, (i) a · d = 0, (ii) a × d = 0, what can be said about the relationship between a and d in each case? (f) Interpret geometrically the vector equation r · n = d, where n is a constant unit vector and d is a constant scalar, stating what d represents. [3] www.MathAcademy.sg 6 © 2019 Math Academy