Converting vector equation of a line to cartesian form
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The document describes how to convert a vector equation of a line to Cartesian form. A vector equation defines a line using a point (a,b,c) on the line and a direction vector (k,m,n). It explains how to write the x, y, z coordinates in terms of λ and rearrange to make λ the subject, then equate the expressions to find the Cartesian equation of the line. The Cartesian equation defines the line using the standard x, y, z variables.
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The document introduces complex numbers and different ways to represent them, including:
1) Imaginary numbers, represented by i, which allows for solutions to equations with "hidden roots". Complex numbers have both a real and imaginary part.
2) Polar form represents complex numbers using modulus (distance from origin) and argument (angle from positive x-axis).
3) Exponential or Euler's form uses modulus and an imaginary exponent to represent complex numbers, where the angle must be in radians.
4) Operations like addition, subtraction, multiplication and division are introduced for complex numbers, along with converting between rectangular, polar and exponential forms.
Lesson 8b the solution of z power n
This document contains three sections about solving complex equations and deriving trigonometric identities using complex numbers. The first section explains how to find the cube root of 2+2i using polar form and de Moivre's Theorem. The second section prompts solving the cube root of 8 + 8i. The third section instructs deriving a formula for cos 4θ in terms of cos θ using binomial expansion and de Moivre's Theorem then equating real and imaginary parts.
Lesson 4 operations with complex numbers in polar form p1 2
The document discusses operations with complex numbers in polar form. It provides the polar form representations of two complex numbers z1 and z2 as r1(cosθ1 + i sin θ1) and r2(cosθ2 + i sin θ2). It asks the reader to find the product and quotient of z1 and z2 using the polar form, and notes that the product can also be expressed as r1 cisθ1 × r2 cisθ2. It provides hints about using trigonometric addition formulas to simplify the expressions and concludes with two practice problems asking the reader to find the product and quotient of complex numbers given in polar form.
Lesson 7 de moivre theorem proof by induction
This document outlines the proof by induction of DeMoivre's Theorem. It presents the three steps of an inductive proof: 1) Show the statement is true for the initial value, here n=1; 2) Assume the statement is true for an arbitrary value k; 3) Show that if true for k then it must also be true for k+1, completing the induction proof that the statement is true for all positive integers n.
Lesson 5 interesting properties of complex conjugates p1 2
This document discusses properties of complex conjugates. It lists six properties to prove about complex conjugates: 1) z+z* and zz* are always real. 2) If z=rcisθ then z*=r cis(-θ). 3) (z+w)*=z*+w* . 4) (zw)*=z*w* . 5) (z/w)*=(z*)/(w*) . 6) (zn)*=(z*)n. It also provides two practice problems: (i) find the complex conjugate of z=(1+iw)2 and (ii) find a complex number z such that z=3z+2z*=5+2i.
Lesson 3 argument polar form of a complex number
This document discusses representing complex numbers in polar form using modulus (r) and argument (θ) instead of Cartesian coordinates (x, y). It shows how to convert between polar and Cartesian forms, defines the notation cis(θ) for representing complex numbers in polar form, and lists some properties of cis(θ) including rules for addition, multiplication, and equivalence under modulo 2π.
Lesson 1 imaginary and complex numbers p1 8
1. Imaginary and complex numbers can be used to solve quadratic equations that have no real solutions. The square root of -1 is defined as i, and complex numbers have both a real and imaginary part.
2. An Argand diagram represents complex numbers graphically by plotting the real part on the x-axis and the imaginary part on a perpendicular y-axis. Common complex number operations like addition, subtraction, multiplication, and division can be performed by treating complex numbers as vectors.
3. The modulus of a complex number z = a + bi is the distance from the point (a, b) to the origin on the Argand diagram, and represents the absolute value of z. The argument of z,
Finding the magnitude of a vector in 2 d and 3d
To find the magnitude (length) of a vector in 2D or 3D, one uses the formula that takes the square root of the sum of the squares of the vector's components. For a vector a with components a1, a2, and a3, the magnitude is calculated as the square root of a1^2 + a2^2 + a3^2. The magnitude of a vector is denoted by |a|.
More scalar product
The document discusses scalar products of vectors and different types of vector relationships. It asks to define the scalar product using notation, calculate products of perpendicular and parallel vectors, and explain the difference between parallel and coincident vectors with a diagram. Perpendicular vectors have a scalar product of 0, parallel vectors maintain the same direction but their scalar product depends on length and angle, and coincident vectors overlap completely and have the greatest possible scalar product.
The scalar product form for the equation of a plane
The document discusses deriving the scalar product form of the equation of a plane. It defines two points A and P on the plane and their position vectors a and r. It explains that the vector from A to P, AP, is equal to r - a. Using the scalar product of the position vector r and the normal vector n to the plane, along with the position vector a, allows obtaining the scalar product equation of the plane, r·n = a·n. This equation can then be written in Cartesian form.
Geometrical interpretation of the magnitude of the cross product
The document discusses the geometric interpretation of the magnitude of the cross product. It explains that the magnitude of the cross product equals the area of the parallelogram formed by the two vectors. Specifically, it describes dropping a perpendicular from one vector to form a right triangle, then using the sine ratio to find the opposite side and thereby the area of the parallelogram. This area is equal to the magnitude of the cross product. It also notes that the cross product can be used to find the area of a triangle.
Properties of the vector product perpendicular and parallel vectors
The document discusses properties of the vector product. It asks the reader to determine if a×b = b×a, ka ×b = k a×b, and a× b + c = a×b + a×c are true by examining numerical examples or vectors a = x! i + y! j + z! k and b = x! i + y! j + z! k. The reader must also justify and explain when a×b = 0, when a×b = a b, when two vectors are parallel, and when two vectors are perpendicular.
Properties of the scalar product perpendicular and parallel vectors
The document discusses properties of the scalar product and asks the reader to determine if several properties are true or not by providing numerical examples or using vector representations. It also asks the reader to justify and explain in words several additional properties related to when the scalar product equals specific values or what defines parallel and perpendicular vectors.
Scalar products structured
This document discusses the scalar product of vectors. It begins by providing an example calculation of the scalar product of two vectors. It then explains that vectors have two basic properties: magnitude and direction. The document instructs the reader to explore how changing the magnitude and direction of vectors affects their scalar product. It poses a series of questions about how the scalar product varies with the angle between vectors and when it is positive, negative, zero, or at its largest/smallest. The summary concludes by asking the reader to generalize how the scalar product relates to the magnitude and direction of vectors.
Y=cosx
The document discusses using the Layar App to access resources related to the Cosine Curve. The Layar App allows users to view augmented reality layers over the real world and likely provides information about mathematical functions or 3D visualizations related to the Cosine Curve. In just a few sentences, the essential idea is that the Layar App serves as a tool to explore augmented reality content about the Cosine Curve function.
Trig radians treasure_hunt
The document contains 10 math problems asking for the smallest or largest value of x for which various trigonometric functions are equal to given values over specified intervals. Each problem is labeled as a card within two sets.
Infographic student project
This document provides information and guidelines for creating an infographic to represent collected data. It lists free websites that can be used to design infographics and recommends including charts to visually depict the data, analyzed headlines, and a journal writing about the data collection and design process. An example infographic about Ted talks is also linked.
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Lesson 4 operations with complex numbers in polar form p1 2
Lesson 4 operations with complex numbers in polar form p1 2
Lesson 5 interesting properties of complex conjugates p1 2
Lesson 5 interesting properties of complex conjugates p1 2
The scalar product form for the equation of a plane
The scalar product form for the equation of a plane
Geometrical interpretation of the magnitude of the cross product
Geometrical interpretation of the magnitude of the cross product
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Properties of the vector product perpendicular and parallel vectors
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