1. The document provides solutions to concepts involving vector addition and subtraction. It includes calculating the magnitude and direction of resultant vectors given the magnitudes and angles between individual vectors.
2. Several problems involve finding components of vectors, angles between vectors, and using trigonometric functions to solve for unknowns.
3. Graphical representations are provided to illustrate concepts such as the independence of the cross product of a velocity vector and position vector from the position of a particle moving along a straight line.
Physics Notes: Solved numerical of Physics first yearRam Chand
1. The document is a physics textbook covering solved numerical problems for the Sindh Textbook Board.
2. It was written by Dr. Ram Chand Raguel and covers topics like scalars and vectors, motion, statics, gravitation, and optics.
3. The author has visited research institutions in the US, Malaysia, Italy, and China and is a member of the American Association of Physics Teachers.
The law of sines, also known as the sine rule, relates the ratios of sides and opposite angles in any triangle. Given any two elements of a triangle (side or angle), the law of sines can be used to calculate the remaining unknown elements. The formula is a/sinA = b/sinB = c/sinC, where a, b, c are the sides and A, B, C are the opposite angles. The document provides examples of using the law of sines to solve for unknown sides and angles in various triangles. It also includes practice problems for students to work through applying the law of sines.
Hilda D. Dragon's least mastered skills are inscribed angles and intercepted arcs. The document defines inscribed angles as angles whose vertex is on a circle and whose sides contain chords of the circle. It defines intercepted arcs as arcs within an inscribed angle that have endpoints on the angle. The document then presents several theorems about inscribed angles and intercepted arcs, such as the measure of an inscribed angle being equal to half the measure of its intercepted arc. It includes example problems and activities for the learner to practice identifying, measuring, and applying theorems about inscribed angles and intercepted arcs.
This document discusses inscribed angles, which are angles whose vertex is on a circle and whose sides contain chords of the circle. It defines key terms like intercepted arc and subtend. Examples are provided to illustrate how to find the measures of inscribed angles, arcs, and angles within inscribed triangles and quadrilaterals using properties of inscribed angles. Students are then given practice problems to solve.
This document provides information on right triangle trigonometry including definitions of basic angle types, right triangle properties, the Pythagorean theorem, trigonometric ratios, and how to solve right triangle problems. It defines trigonometric functions like sine, cosine, and tangent in terms of an acute angle and adjacent/opposite sides. Examples are given for finding missing side lengths and converting between angle units. Practice problems apply the concepts to evaluate trig functions and solve application problems involving heights, distances, and angles of elevation/depression.
1) The document discusses vector addition of forces using the parallelogram law and trigonometry.
2) Forces can be added by constructing a parallelogram with the force vectors as sides and the resultant vector as the diagonal.
3) Trigonometric relationships like the law of sines and cosines allow determining the magnitudes and directions of resultant and component forces.
Physics Notes: Solved numerical of Physics first yearRam Chand
1. The document is a physics textbook covering solved numerical problems for the Sindh Textbook Board.
2. It was written by Dr. Ram Chand Raguel and covers topics like scalars and vectors, motion, statics, gravitation, and optics.
3. The author has visited research institutions in the US, Malaysia, Italy, and China and is a member of the American Association of Physics Teachers.
The law of sines, also known as the sine rule, relates the ratios of sides and opposite angles in any triangle. Given any two elements of a triangle (side or angle), the law of sines can be used to calculate the remaining unknown elements. The formula is a/sinA = b/sinB = c/sinC, where a, b, c are the sides and A, B, C are the opposite angles. The document provides examples of using the law of sines to solve for unknown sides and angles in various triangles. It also includes practice problems for students to work through applying the law of sines.
Hilda D. Dragon's least mastered skills are inscribed angles and intercepted arcs. The document defines inscribed angles as angles whose vertex is on a circle and whose sides contain chords of the circle. It defines intercepted arcs as arcs within an inscribed angle that have endpoints on the angle. The document then presents several theorems about inscribed angles and intercepted arcs, such as the measure of an inscribed angle being equal to half the measure of its intercepted arc. It includes example problems and activities for the learner to practice identifying, measuring, and applying theorems about inscribed angles and intercepted arcs.
This document discusses inscribed angles, which are angles whose vertex is on a circle and whose sides contain chords of the circle. It defines key terms like intercepted arc and subtend. Examples are provided to illustrate how to find the measures of inscribed angles, arcs, and angles within inscribed triangles and quadrilaterals using properties of inscribed angles. Students are then given practice problems to solve.
This document provides information on right triangle trigonometry including definitions of basic angle types, right triangle properties, the Pythagorean theorem, trigonometric ratios, and how to solve right triangle problems. It defines trigonometric functions like sine, cosine, and tangent in terms of an acute angle and adjacent/opposite sides. Examples are given for finding missing side lengths and converting between angle units. Practice problems apply the concepts to evaluate trig functions and solve application problems involving heights, distances, and angles of elevation/depression.
1) The document discusses vector addition of forces using the parallelogram law and trigonometry.
2) Forces can be added by constructing a parallelogram with the force vectors as sides and the resultant vector as the diagonal.
3) Trigonometric relationships like the law of sines and cosines allow determining the magnitudes and directions of resultant and component forces.
This document contains partial solutions to homework problems from dynamics courses taught between 2002-2003. It was compiled by the author from PDF files to help recipients with their studies. It includes solutions for chapters 13-17 and past exams, but is not a complete set of answers. The mass, pulley, and rod problems solved here provide example solutions that could aid readers in learning concepts in engineering dynamics.
This document provides examples and explanations of concepts related to tangents of circles. It begins with identifying lines and segments that intersect circles, such as chords, secants, tangents, diameters, and radii. Several examples are worked through that find the radius of circles and identify points of tangency and equations of tangent lines. The document also covers common tangents to two circles and using properties of tangents to solve problems. Application examples include finding the distance from a spacecraft to Earth's horizon and the summit of a mountain to the horizon.
This document discusses properties of arcs and chords in circles. It begins with objectives and vocabulary definitions for arcs, central angles, minor arcs, major arcs, and adjacent arcs. Examples are then provided to illustrate finding measures of arcs and angles using properties such as the arc addition postulate and that congruent arcs have congruent chords. Further examples apply properties to find measures of arcs, angles, and chords in various circle graphs and diagrams. Practice problems are also included for students to check their understanding.
This document provides examples and explanations of how to find angle measures formed by lines intersecting within and on circles. It includes examples of finding angle measures using tangent-secant angles, tangent-chord angles, and angles inside and on circles. Students are guided through step-by-step workings and are given practice problems to solve involving finding specific angle measures using the concepts taught.
1) The document discusses resolving forces into components that are parallel and perpendicular to inclined planes. It shows examples of resolving weights and tensions into these components.
2) Key concepts covered include resolving forces, balancing horizontal and vertical forces, and calculating tensions, normal forces, and coefficients of friction.
3) Several multi-step examples are worked through that demonstrate resolving forces for objects on inclined planes and calculating unknown values like tensions and coefficients of friction.
This presentation covers scalar quantity, vector quantity, addition of vectors & multiplication of vector. I hope this PPT will be helpful for Instructors as well as students.
The document contains solutions to examples from chapters in a strength of materials textbook, solving for things like beam reactions, stresses, deflections, shear and moment diagrams. It analyzes beams, braces, and steel strips under different loads and conditions to calculate values like forces, stresses, moments and radii of curvature. Diagrams are included to illustrate the shear and moment diagrams for one of the beam examples.
The document presents solutions to statics problems involving levers and free body diagrams. It includes the following steps:
1) Drawing a free body diagram of the lever system showing applied forces.
2) Applying equations of equilibrium to solve for unknown reaction forces, including summing moments about a point equal to zero and summing horizontal and vertical forces equal to zero.
3) Solving the equations to find the values of the unknown reaction forces acting on the lever system.
1. The document provides examples of unit conversions using metric prefixes and other conversion factors. Calculations are shown for distances, areas, volumes, and time conversions between various units like kilometers, meters, centimeters, microseconds, seconds and years.
2. Geometric formulas are used to calculate properties of the Earth like circumference, surface area and volume. Examples of unit conversions involving length, area, volume, time and other units are also provided.
3. Examples involve calculating distances, areas, volumes, rotations, uncertainties and time differences between various measurement systems including metric, English, and other historical units. Careful application of conversion factors and significant figures is emphasized.
This document discusses scale in aerial photography and provides examples of calculating scale and determining flying height from a vertical photograph. It contains the following key points:
1) Scale in aerial photography is the ratio of the distance between two points on an image to the actual distance on the ground, and it varies depending on the terrain and altitude of the airplane.
2) Examples are provided to demonstrate calculating scale based on given focal length, altitude, and point elevations. Flying height is calculated using similar triangles relationships between photo and ground distances.
3) A numerical example shows determining the flying height that provides a measured photo distance equal to the known ground distance, through an iterative approximation process.
The document discusses circles in the coordinate plane. It provides examples of writing the equation of a circle given its center and radius or given two points it passes through. It also gives examples of graphing circles given their equations by identifying the center and radius. Additionally, it presents an example of using three points and their perpendicular bisectors to find the center of a circle passing through those three points, and applies this to solving a problem about finding the location of a structure.
This module discusses solving oblique triangles using the law of sines. It begins by introducing acute and obtuse triangles and how to find the measure of the third angle given two angles. It then derives the law of sines and shows how it can be used to solve triangles where two angles and a side opposite one angle are given, two angles and the included side are given, or all three sides are given. Examples of solving various triangle scenarios are provided.
This document provides information about solving triangles using the Law of Sines. It begins with an introduction to the Law of Sines and how it relates trigonometric ratios of the angles to the sides of a triangle. It then presents examples of using the Law of Sines to solve triangles given: two angles and a side opposite one angle; two angles and the included side; and two sides and an angle opposite one side. Several practice problem sets are provided for each case. The document aims to teach readers how to apply the Law of Sines to solve different types of triangle problems.
Structural Analysis (Solutions) Chapter 9 by WajahatWajahat Ullah
The document provides information about determining displacements of joints in truss structures using the method of virtual work and Castigliano's theorem. It includes the geometry, applied forces, and cross-sectional areas of sample truss problems. The user is asked to determine the vertical displacement of various joints by calculating the internal virtual work of the truss members. Solutions are provided using both the virtual work method and Castigliano's theorem.
This document discusses using theorems about circle segment lengths to solve problems. It provides examples of applying the chord-chord product theorem, secant-secant product theorem, and secant-tangent product theorem to find missing segment lengths and values of variables. The examples include finding diameters of disks and plates from archeological discoveries, constructing a wooden moon prop, and solving geometry problems involving circles. Students are then quizzed on additional applications of these theorems to find lengths and values.
Chapter 1 lines and angles ii [compatibility mode]Khusaini Majid
1. The document defines key properties of angles associated with parallel lines and transversals, including corresponding angles, alternate angles, and interior angles.
2. It provides examples of using these properties to determine if lines are parallel and calculate angle measures.
3. Exercises at the end provide additional practice problems involving parallel lines and their associated angles.
The document discusses trigonometric graphs and their key characteristics. It provides examples of trigonometric functions and shows how to determine the period, amplitude, and phase shift of sine and cosine graphs. The period is 2π/b, the amplitude is the absolute value of a, and the phase shift is the absolute value of c, which shifts the graph right or left depending on whether c is positive or negative. Examples are worked through step-by-step to illustrate how to find these values and sketch the graph of one period.
Copier correction du devoir_de_synthèse_de_topographieAhmed Manai
1. This document contains the steps to solve two topography exercises.
2. The first exercise involves determining angles, distances, and areas of triangles between stations A, B, C, and D using trigonometric relationships and angle measurements.
3. The second exercise calculates the horizontal distance between station S and a building using zenith angle measurements and the distance measured on a staff. It then determines the height of the building.
Este documento resume brevemente la historia institucional y personal de una persona. Describe cómo durante dos años todo sucedió normalmente en la escuela, pero algo cambió. Luego detalla un evento en el que se suspendió una fiesta de la expresión, lo que provocó una explosión de rabia. Finalmente, la persona visitó a un psicólogo y dio explicaciones a cada curso sobre lo sucedido, disculpándose. Hasta el día de hoy, los alumnos de ese turno nocturno no tienen más fiestas de la expresión.
Este documento resume brevemente la historia institucional y personal de una persona. Describe cómo durante dos años todo sucedió normalmente en la escuela, pero algo cambió. Luego detalla un evento en el que se suspendió una fiesta de la expresión, lo que provocó un estallido de rabia. Finalmente, la persona visitó a un psicólogo y comprendió que tenía fobia al vicedirector de la escuela. Desde entonces, los estudiantes de ese turno nocturno no han vuelto a tener fiestas de la expresión.
This document contains partial solutions to homework problems from dynamics courses taught between 2002-2003. It was compiled by the author from PDF files to help recipients with their studies. It includes solutions for chapters 13-17 and past exams, but is not a complete set of answers. The mass, pulley, and rod problems solved here provide example solutions that could aid readers in learning concepts in engineering dynamics.
This document provides examples and explanations of concepts related to tangents of circles. It begins with identifying lines and segments that intersect circles, such as chords, secants, tangents, diameters, and radii. Several examples are worked through that find the radius of circles and identify points of tangency and equations of tangent lines. The document also covers common tangents to two circles and using properties of tangents to solve problems. Application examples include finding the distance from a spacecraft to Earth's horizon and the summit of a mountain to the horizon.
This document discusses properties of arcs and chords in circles. It begins with objectives and vocabulary definitions for arcs, central angles, minor arcs, major arcs, and adjacent arcs. Examples are then provided to illustrate finding measures of arcs and angles using properties such as the arc addition postulate and that congruent arcs have congruent chords. Further examples apply properties to find measures of arcs, angles, and chords in various circle graphs and diagrams. Practice problems are also included for students to check their understanding.
This document provides examples and explanations of how to find angle measures formed by lines intersecting within and on circles. It includes examples of finding angle measures using tangent-secant angles, tangent-chord angles, and angles inside and on circles. Students are guided through step-by-step workings and are given practice problems to solve involving finding specific angle measures using the concepts taught.
1) The document discusses resolving forces into components that are parallel and perpendicular to inclined planes. It shows examples of resolving weights and tensions into these components.
2) Key concepts covered include resolving forces, balancing horizontal and vertical forces, and calculating tensions, normal forces, and coefficients of friction.
3) Several multi-step examples are worked through that demonstrate resolving forces for objects on inclined planes and calculating unknown values like tensions and coefficients of friction.
This presentation covers scalar quantity, vector quantity, addition of vectors & multiplication of vector. I hope this PPT will be helpful for Instructors as well as students.
The document contains solutions to examples from chapters in a strength of materials textbook, solving for things like beam reactions, stresses, deflections, shear and moment diagrams. It analyzes beams, braces, and steel strips under different loads and conditions to calculate values like forces, stresses, moments and radii of curvature. Diagrams are included to illustrate the shear and moment diagrams for one of the beam examples.
The document presents solutions to statics problems involving levers and free body diagrams. It includes the following steps:
1) Drawing a free body diagram of the lever system showing applied forces.
2) Applying equations of equilibrium to solve for unknown reaction forces, including summing moments about a point equal to zero and summing horizontal and vertical forces equal to zero.
3) Solving the equations to find the values of the unknown reaction forces acting on the lever system.
1. The document provides examples of unit conversions using metric prefixes and other conversion factors. Calculations are shown for distances, areas, volumes, and time conversions between various units like kilometers, meters, centimeters, microseconds, seconds and years.
2. Geometric formulas are used to calculate properties of the Earth like circumference, surface area and volume. Examples of unit conversions involving length, area, volume, time and other units are also provided.
3. Examples involve calculating distances, areas, volumes, rotations, uncertainties and time differences between various measurement systems including metric, English, and other historical units. Careful application of conversion factors and significant figures is emphasized.
This document discusses scale in aerial photography and provides examples of calculating scale and determining flying height from a vertical photograph. It contains the following key points:
1) Scale in aerial photography is the ratio of the distance between two points on an image to the actual distance on the ground, and it varies depending on the terrain and altitude of the airplane.
2) Examples are provided to demonstrate calculating scale based on given focal length, altitude, and point elevations. Flying height is calculated using similar triangles relationships between photo and ground distances.
3) A numerical example shows determining the flying height that provides a measured photo distance equal to the known ground distance, through an iterative approximation process.
The document discusses circles in the coordinate plane. It provides examples of writing the equation of a circle given its center and radius or given two points it passes through. It also gives examples of graphing circles given their equations by identifying the center and radius. Additionally, it presents an example of using three points and their perpendicular bisectors to find the center of a circle passing through those three points, and applies this to solving a problem about finding the location of a structure.
This module discusses solving oblique triangles using the law of sines. It begins by introducing acute and obtuse triangles and how to find the measure of the third angle given two angles. It then derives the law of sines and shows how it can be used to solve triangles where two angles and a side opposite one angle are given, two angles and the included side are given, or all three sides are given. Examples of solving various triangle scenarios are provided.
This document provides information about solving triangles using the Law of Sines. It begins with an introduction to the Law of Sines and how it relates trigonometric ratios of the angles to the sides of a triangle. It then presents examples of using the Law of Sines to solve triangles given: two angles and a side opposite one angle; two angles and the included side; and two sides and an angle opposite one side. Several practice problem sets are provided for each case. The document aims to teach readers how to apply the Law of Sines to solve different types of triangle problems.
Structural Analysis (Solutions) Chapter 9 by WajahatWajahat Ullah
The document provides information about determining displacements of joints in truss structures using the method of virtual work and Castigliano's theorem. It includes the geometry, applied forces, and cross-sectional areas of sample truss problems. The user is asked to determine the vertical displacement of various joints by calculating the internal virtual work of the truss members. Solutions are provided using both the virtual work method and Castigliano's theorem.
This document discusses using theorems about circle segment lengths to solve problems. It provides examples of applying the chord-chord product theorem, secant-secant product theorem, and secant-tangent product theorem to find missing segment lengths and values of variables. The examples include finding diameters of disks and plates from archeological discoveries, constructing a wooden moon prop, and solving geometry problems involving circles. Students are then quizzed on additional applications of these theorems to find lengths and values.
Chapter 1 lines and angles ii [compatibility mode]Khusaini Majid
1. The document defines key properties of angles associated with parallel lines and transversals, including corresponding angles, alternate angles, and interior angles.
2. It provides examples of using these properties to determine if lines are parallel and calculate angle measures.
3. Exercises at the end provide additional practice problems involving parallel lines and their associated angles.
The document discusses trigonometric graphs and their key characteristics. It provides examples of trigonometric functions and shows how to determine the period, amplitude, and phase shift of sine and cosine graphs. The period is 2π/b, the amplitude is the absolute value of a, and the phase shift is the absolute value of c, which shifts the graph right or left depending on whether c is positive or negative. Examples are worked through step-by-step to illustrate how to find these values and sketch the graph of one period.
Copier correction du devoir_de_synthèse_de_topographieAhmed Manai
1. This document contains the steps to solve two topography exercises.
2. The first exercise involves determining angles, distances, and areas of triangles between stations A, B, C, and D using trigonometric relationships and angle measurements.
3. The second exercise calculates the horizontal distance between station S and a building using zenith angle measurements and the distance measured on a staff. It then determines the height of the building.
Este documento resume brevemente la historia institucional y personal de una persona. Describe cómo durante dos años todo sucedió normalmente en la escuela, pero algo cambió. Luego detalla un evento en el que se suspendió una fiesta de la expresión, lo que provocó una explosión de rabia. Finalmente, la persona visitó a un psicólogo y dio explicaciones a cada curso sobre lo sucedido, disculpándose. Hasta el día de hoy, los alumnos de ese turno nocturno no tienen más fiestas de la expresión.
Este documento resume brevemente la historia institucional y personal de una persona. Describe cómo durante dos años todo sucedió normalmente en la escuela, pero algo cambió. Luego detalla un evento en el que se suspendió una fiesta de la expresión, lo que provocó un estallido de rabia. Finalmente, la persona visitó a un psicólogo y comprendió que tenía fobia al vicedirector de la escuela. Desde entonces, los estudiantes de ese turno nocturno no han vuelto a tener fiestas de la expresión.
La Unión Europea ha acordado un embargo petrolero contra Rusia en respuesta a la invasión de Ucrania. El embargo prohibirá las importaciones marítimas de petróleo ruso a la UE y pondrá fin a las entregas a través de oleoductos dentro de seis meses. Esta medida forma parte de un sexto paquete de sanciones de la UE destinadas a aumentar la presión económica sobre Moscú y privar al Kremlin de fondos para financiar su guerra.
1. The document provides dimensional analysis and units for various physical quantities. It lists 18 physical quantities along with their dimensions and standard SI units.
2. Key physical quantities included are: force, work, power, gravitational constant, angular velocity, angular momentum, moment of inertia, torque, Young's modulus, surface tension, viscosity, pressure, intensity of waves, specific heat capacity, Stefan's constant, thermal conductivity, current density, and electrical conductivity.
3. The dimensional analysis verifies the consistency of equations involving these physical quantities by ensuring the dimensions on both sides of equations are equal. This helps identify any dimensionally incorrect equations.
The document discusses Mansion Barbershop, a barbershop franchise concept in Indonesia. It notes a shifting trend of more men spending time and money on grooming. Mansion aims to be beyond a traditional barbershop, finely crafted for men's needs with a manly ambiance. It offers haircuts, massage, innovative seating and a smart queuing system. Analysis shows the franchise model has a 12.6% ROI and 8 month payback period, while the investor model has a 5.8% ROI and 17 month payback period. Contact details and outlet locations are provided.
1. The document provides examples of converting between different units of measurement using metric prefixes like micro, pico, and nano. Conversions include kilometers to microns, centimeters to microns, and yards to microns.
2. Additional examples convert between inches and picas, points, and other units like grys and almudes which are used to measure volumes of grains.
3. Conversion factors are also provided for seconds, days, weeks, centuries and other units of time. Examples show converting between these different units of time.
1. The question involves calculating the resultant of two vectors A and B. The angle between the vectors is 90 degrees. The resultant vector R makes an angle of 73 degrees with the x-axis.
2. The question again involves calculating the resultant R of two vectors A and B. The angle between the vectors is 30 degrees. Both vectors have a magnitude of 10 units. The magnitude of the resultant vector R is calculated to be 19.3 units.
1) The document contains examples calculating various vector operations such as finding unit vectors, magnitudes, dot and cross products, and vector components.
2) It also contains examples finding vector fields, surfaces where vector field components are equal to scalars, and demonstrating properties of vector fields such as being everywhere parallel.
3) The document tests understanding of vector concepts through multiple practice problems.
1. The document analyzes the forces acting on a rope as it falls over a support. It derives an expression for the force N from the support as a function of time t until the free end of the rope reaches a distance of 2L.
2. It examines the motion of a mass attached to a string winding around a thin pole, and derives an expression for the angle θ at which the string becomes fully unwound using conservation of energy and the horizontal component of tension.
3. It considers electrical resistance in a circuit composed of squares connected at the corners. By simplifying the circuit in stages, it derives an expression for the desired resistance factor x between opposite corners of the original circuit.
1. The document analyzes the forces acting on a rope as it falls over a support. It derives an expression for the force N from the support as a function of time t until the free end of the rope falls a distance of 2L.
2. It examines the motion of a mass attached to a string winding around a thin pole, and derives an expression for the angle θ that the string will make with the horizontal when fully unwound.
3. It considers electrical resistance in a circuit made of squares connected at the corners. It derives an expression for the factor x that determines the desired resistance between opposite corners.
This document provides an outline and answers for a chapter on vectors. It begins with an outline of topics to be covered, including coordinate systems, vector and scalar quantities, properties of vectors, and components of a vector. It then provides answers to various questions related to the chapter topics, such as determining whether quantities are vectors or scalars, calculating displacements and distances, adding and subtracting vectors, and finding vector components.
This document provides an outline and answers for a chapter on vectors. It begins with an outline of topics to be covered, including coordinate systems, vector and scalar quantities, properties of vectors, and components of a vector. It then provides answers to various questions related to the chapter topics, such as determining whether quantities are vectors or scalars, calculating displacements and distances, adding and subtracting vectors graphically, and finding vector components. Solutions to example problems are also given for additional concepts like unit vectors, vector addition, and determining vector magnitudes and directions.
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.
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The dot product of two vectors A and B, written as A·B, is a scalar equal to the magnitude of A times the magnitude of B times the cosine of the angle between them. The dot product is commutative and distributes over vector addition. It can be used to find the component of a vector that is parallel to a given line and the angle between two vectors. Applications include finding the angle between two lines and resolving a force into components parallel and perpendicular to a structural member.
This document provides an overview of trigonometry including plane and spherical trigonometry. It covers topics such as angle measurements, the six trigonometric functions, solving right triangles using the Pythagorean theorem, solving oblique triangles using laws of sines and cosines, inverse trigonometric functions, trigonometric identities, and area of triangles. It also includes sample problems and their solutions related to these topics.
The document contains multiple math word problems and their step-by-step solutions. It asks the reader to calculate various values like interest earned over time, the measures of angles, lengths and volumes of geometric shapes, rates of work, and more. The problems are explained clearly with diagrams and algebraic equations to show the rationalization for arriving at the answers.
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.
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.
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.
1. 2.1
SOLUTIONS TO CONCEPTS
CHAPTER – 2
1. As shown in the figure,
The angle between A
and B
= 110° – 20° = 90°
|A|
= 3 and |B|
= 4m
Resultant R = cosAB2BA 22
= 5 m
Let be the angle between R
and A
=
90cos43
90sin4
tan 1
= tan
–1
(4/3) = 53°
Resultant vector makes angle (53° + 20°) = 73° with x-axis.
2. Angle between A
and B
is = 60° – 30° =30°
| A
| and |B
| = 10 unit
R = 2 2
10 10 2.10.10.cos30 = 19.3
be the angle between R
and A
= tan
–1 110sin30 1
tan
10 10cos30 2 3
= tan
–1
(0.26795) = 15°
Resultant makes 15° + 30° = 45° angle with x-axis.
3. x component of A
= 100 cos 45° = 2/100 unit
x component of B
= 100 cos 135° = 2/100
x component of C
= 100 cos 315° = 2/100
Resultant x component = 2/100 – 2/100 + 2/100 = 2/100
y component of A
= 100 sin 45° = 2/100 unit
y component of B
= 100 sin 135° = 2/100
y component of C
= 100 sin 315° = – 2/100
Resultant y component = 2/100 + 2/100 – 2/100 = 2/100
Resultant = 100
Tan =
componentx
componenty
= 1
= tan
–1
(1) = 45°
The resultant is 100 unit at 45° with x-axis.
4. j3i4a
, j4i3b
a) 22
34|a|
= 5
b) 169|b|
= 5
c) 27|j7i7||ba|
d) ˆ ˆ ˆ ˆa b ( 3 4)i ( 4 3)j i j
2 2
| a b | 1 ( 1) 2
.
x
y
R
B
A
20
x
y
B
A
30°
60°
315°
45°
135°
2. Chapter-2
2.2
5. x component of OA = 2cos30° = 3
x component of BC = 1.5 cos 120° = –0.75
x component of DE = 1 cos 270° = 0
y component of OA = 2 sin 30° = 1
y component of BC = 1.5 sin 120° = 1.3
y component of DE = 1 sin 270° = –1
Rx = x component of resultant = 075.03 = 0.98 m
Ry = resultant y component = 1 + 1.3 – 1 = 1.3 m
So, R = Resultant = 1.6 m
If it makes and angle with positive x-axis
Tan =
componentx
componenty
= 1.32
= tan
–1
1.32
6. |a|
= 3m |b|
= 4
a) If R = 1 unit cos.4.3.243 22
= 1
= 180°
b) cos.4.3.243 22
= 5
= 90°
c) cos.4.3.243 22
= 7
= 0°
Angle between them is 0°.
7. Kˆ4Jˆ5.0iˆ2AD = ˆ ˆ6i 0.5j
AD = 22
DEAE = 6.02 KM
Tan = DE / AE = 1/12
= tan
–1
(1/12)
The displacement of the car is 6.02 km along the distance tan
–1
(1/12) with positive x-axis.
8. In ABC, tan = x/2 and in DCE, tan = (2 – x)/4 tan = (x/2) = (2 – x)/4 = 4x
4 – 2x = 4x
6x = 4 x = 2/3 ft
a) In ABC, AC = 22
BCAB =
2
10
3
ft
b) In CDE, DE = 1 – (2/3) = 4/3 ft
CD = 4 ft. So, CE = 22
DECD =
4
10
3
ft
c) In AGE, AE = 22
GEAG = 2 2 ft.
9. Here the displacement vector kˆ3jˆ4iˆ7r
a) magnitude of displacement = 74 ft
b) the components of the displacement vector are 7 ft, 4 ft and 3 ft.
2m
D
A
E
B
x
O
y
1m
1.5m
90°
30° 60°
6m
EA
D
B
0.5 km
C
2m
4m
0.5 km
2–x
G
A
D
B BC = 2 ft
AF = 2 ft
DE = 2x
x
C
E
F
r
z
Y
3. Chapter-2
2.3
10. a
is a vector of magnitude 4.5 unit due north.
a) 3| a
| = 3 4.5 = 13.5
3a
is along north having magnitude 13.5 units.
b) –4| a
| = –4 1.5 = –6 unit
–4 a
is a vector of magnitude 6 unit due south.
11. | a
| = 2 m, | b
| = 3 m
angle between them = 60°
a) 60cos|b||a|ba
= 2 3 1/2 = 3 m
2
b) 60sin|b||a||ba|
= 2 3 3/ 2 = 3 3 m
2
.
12. We know that according to polygon law of vector addition, the resultant
of these six vectors is zero.
Here A = B = C = D = E = F (magnitude)
So, Rx = A cos + A cos /3 + A cos 2/3 + A cos 3/3 + A cos 4/4 +
A cos 5/5 = 0
[As resultant is zero. X component of resultant Rx = 0]
= cos + cos /3 + cos 2/3 + cos 3/3 + cos 4/3 + cos 5/3 = 0
Note : Similarly it can be proved that,
sin + sin /3 + sin 2/3 + sin 3/3 + sin 4/3 + sin 5/3 = 0
13. a 2i 3 j 4k; b 3i 4 j 5k
cosabba
ab
ba
cos 1
1 1
2 2 2 2 2 2
2 3 3 4 4 5 38
cos cos
14502 3 4 3 4 5
14. 0)BA(A
(claim)
As, nˆsinABBA
AB sin nˆ is a vector which is perpendicular to the plane containing A
and B
, this implies that it is
also perpendicular to A
. As dot product of two perpendicular vector is zero.
Thus 0)BA(A
.
15. ˆ ˆ ˆA 2i 3j 4k
, ˆ ˆ ˆB 4i 3j 2k
ˆ ˆ ˆi j k
A B 2 3 4
4 3 2
ˆ ˆ ˆ ˆ ˆ ˆi(6 12) j(4 16) k(6 12) 6i 12j 6k .
16. Given that A
, B
and C
are mutually perpendicular
A
× B
is a vector which direction is perpendicular to the plane containing A
and B
.
Also C
is perpendicular to A
and B
Angle between C
and A
× B
is 0° or 180° (fig.1)
So, C
× ( A
× B
) = 0
The converse is not true.
For example, if two of the vector are parallel, (fig.2), then also
C
× ( A
× B
) = 0
So, they need not be mutually perpendicular.
A1
60° = /3
A2
A3
A4A5
A6
(A B)
A
B
C
A
B
C
4. Chapter-2
2.4
17. The particle moves on the straight line PP’ at speed v.
From the figure,
nˆsinv)OP(vOP = v(OP) sin ˆn = v(OQ) ˆn
It can be seen from the figure, OQ = OP sin = OP’ sin ’
So, whatever may be the position of the particle, the magnitude and
direction of vOP
remain constant.
vOP
is independent of the position P.
18. Give 0)Bv(qEqF
)Bv(E
So, the direction of Bv
should be opposite to the direction of E
. Hence,
v
should be in the positive yz-plane.
Again, E = vB sin v =
sinB
E
For v to be minimum, = 90° and so vmin = F/B
So, the particle must be projected at a minimum speed of E/B along +ve z-axis ( = 90°) as shown in the
figure, so that the force is zero.
19. For example, as shown in the figure,
A B
B
along west
B C
A
along south
C
along north
A B
= 0 A B B C
B C
= 0 But B C
20. The graph y = 2x
2
should be drawn by the student on a graph paper for exact
results.
To find slope at any point, draw a tangent at the point and extend the line to meet
x-axis. Then find tan as shown in the figure.
It can be checked that,
Slope = tan = )x2(
dx
d
dx
dy 2
= 4x
Where x = the x-coordinate of the point where the slope is to be measured.
21. y = sinx
So, y + y = sin (x + x)
y = sin (x + x) – sin x
= sin
3 100 3
= 0.0157.
22. Given that, i = RC/t
0ei
Rate of change of current = RC/t
0
RC/i
0 e
dt
d
iei
dt
d
dt
di
= t / RC0i
e
RC
When a) t = 0,
RC
i
dt
di
b) when t = RC,
RCe
i
dt
di
c) when t = 10 RC, 10
0
RCe
i
dt
di
V
Q
O
P P
x
E
B
V
y
C
B
A
B
x
y=2x2 y
x
y = sinx
y
5. Chapter-2
2.5
23. Equation i = RC/t
0ei
i0 = 2A, R = 6 10
–5
, C = 0.0500 10
–6
F = 5 10
–7
F
a) i =
3 7
0.3 0.3
6 0 5 10 0.3 2
2 e 2 e amp
e
.
b) t / RC0idi
e
dt RC
when t = 0.3 sec ( 0.3 / 0.3)di 2 20
e Amp/ sec
dt 0.30 3e
c) At t = 0.31 sec, i = ( 0.3 / 0.3) 5.8
2e Amp
3e
.
24. y = 3x
2
+ 6x + 7
Area bounded by the curve, x axis with coordinates with x = 5 and x = 10 is
given by,
Area =
y
0
dy =
10
2
5
(3x 6x 7)dx =
10 103 2
10
5
5 5
x x
3 5 7x
3 3
= 1135 sq.units.
25. Area =
y
0
dy = 0
0
sinxdx [cosx]
= 2
26. The given function is y = e
–x
When x = 0, y = e
–0
= 1
x increases, y value deceases and only at x = , y = 0.
So, the required area can be found out by integrating the function from 0 to .
So, Area = x x
0
0
e dx [e ] 1
.
27. bxa
length
mass
a) S.I. unit of ‘a’ = kg/m and SI unit of ‘b’ = kg/m
2
(from principle of
homogeneity of dimensions)
b) Let us consider a small element of length ‘dx’ at a distance x from the
origin as shown in the figure.
dm = mass of the element = dx = (a + bx) dx
So, mass of the rod = m = dx)bxa(dm
L
0
=
L2 2
0
bx bL
ax aL
2 2
28.
dp
dt
= (10 N) + (2 N/S)t
momentum is zero at t = 0
momentum at t = 10 sec will be
dp = [(10 N) + 2Ns t]dt
p 10 10
0 0 0
dp 10dt (2tdt) =
102
10
0
0
t
10t 2
2
= 200 kg m/s.
5
y = 3x2
+ 6x + 7
10
x
y
y = sinx
y
x
y
x
x =1
O
y
x
6. Chapter-2
2.6
29. The change in a function of y and the independent variable x are related as 2
x
dx
dy
.
dy = x
2
dx
Taking integration of both sides,
2
dy x dx y =
3
x
c
3
y as a function of x is represented by y =
3
x
c
3
.
30. The number significant digits
a) 1001 No.of significant digits = 4
b) 100.1 No.of significant digits = 4
c) 100.10 No.of significant digits = 5
d) 0.001001 No.of significant digits = 4
31. The metre scale is graduated at every millimeter.
1 m = 100 mm
The minimum no.of significant digit may be 1 (e.g. for measurements like 5 mm, 7 mm etc) and the
maximum no.of significant digits may be 4 (e.g.1000 mm)
So, the no.of significant digits may be 1, 2, 3 or 4.
32. a) In the value 3472, after the digit 4, 7 is present. Its value is greater than 5.
So, the next two digits are neglected and the value of 4 is increased by 1.
value becomes 3500
b) value = 84
c) 2.6
d) value is 28.
33. Given that, for the cylinder
Length = l = 4.54 cm, radius = r = 1.75 cm
Volume = r
2
l = (4.54) (1.75)
2
Since, the minimum no.of significant digits on a particular term is 3, the result should have
3 significant digits and others rounded off.
So, volume V = r
2
l = (3.14) (1.75) (1.75) (4.54) = 43.6577 cm
3
Since, it is to be rounded off to 3 significant digits, V = 43.7 cm
3
.
34. We know that,
Average thickness =
2.17 2.17 2.18
3
= 2.1733 mm
Rounding off to 3 significant digits, average thickness = 2.17 mm.
35. As shown in the figure,
Actual effective length = (90.0 + 2.13) cm
But, in the measurement 90.0 cm, the no. of significant digits is only 2.
So, the addition must be done by considering only 2 significant digits of each
measurement.
So, effective length = 90.0 + 2.1 = 92.1 cm.
* * * *
r
l
90cm
2.13cm