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Slide 2 / 159 Pre-Calculus Vectors 2015-03-24 www.njctl.org
Slide 3 / 159 Table of Contents Intro to Vectors Converting Rectangular and Polar Forms Operations with Vectors Scalar Multiples Addition Subtraction Vector Equations of Lines Dot Product Angle Between Vectors 3-Dimensional Space Vectors, Lines, and Planes
Slide 4 / 159 Intro to Vectors Return to Table of Contents
Slide 5 / 159 Intro to Vectors Drawing A Vector A vector is always drawn with an arrow at the tip indicating the direction, and the length of the line determines the magnitude. Remember displacement is the distance away from your initial position, it does not account for the actual distance you moved
Slide 6 / 159 Intro to Vectors Determining magnitude and direction anti-parallel All of these vectors have the same magnitude, but vector B runs anti-parallel therefore it is denoted negative A.
Slide 7 / 159 Intro to Vectors Draw a vector to represent: a plane flying North-East at 500 mph a wind blowing to the west at 50 mph
Slide 8 / 159 Intro to Vectors Draw a vector to represent: a boat traveling west at 4 knots a current traveling south at 2 knots
Slide 9 / 159 Intro to Vectors 1 Which vector represents a car driving east at 60 mph? A C B D
Slide 10 / 159 Intro to Vectors 2 Which vector represents a wind blowing south at 30 mph if the given vector represents a car driving east at 60mph? A B C D
Slide 11 / 159 Intro to Vectors 3 A rabbit runs east 30 feet and then north 40 feet, which vector represents the rabbits displacement from its starting position? A B C D
Slide 12 / 159 Intro to Vectors There are 2 kinds of vectors drawn on a coordinate grid: Those that start Those that don't from the origin: start at the origin: joins (2, -4) to (4, 4)
Slide 13 / 159 Intro to Vectors A vector can be broken down into its component forces. u x is the horizontal part. u y is the vertical part. In u x = 8 u y = 6
Slide 14 / 159 Intro to Vectors joins (2, -4) to (4, 4)
Slide 15 / 159 Intro to Vectors 4 Which vector is in standard position? A B C D
Slide 16 / 159 Intro to Vectors 5 What is?
Slide 17 / 159 Intro to Vectors 6 What is?
Slide 18 / 159 Intro to Vectors 7 What is?
Slide 19 / 159 Intro to Vectors 8 What is?
Slide 20 / 159 Intro to Vectors 9 What is?
Slide 21 / 159 Intro to Vectors 10 What is?
Slide 22 / 159 Intro to Vectors 11 What is?
Slide 23 / 159 Intro to Vectors 12 What is?
Slide 24 / 159 Intro to Vectors After a vector is broken down into its component forces,the magnitude of the vector can be calculated. = magnitude (length) of u x is the horizontal part. u y is the vertical part. In u x = 8 u y = 6 6 8
Slide 25 / 159 Intro to Vectors A plane is traveling with an eastern component of 40 miles and a northern component of 75 miles, every hour. Draw a representation of this, include the vector of the planes actual path. What is the planes displacement traveled after one hour?
Slide 26 / 159 Intro to Vectors 13 If a car under goes a displacement of 3 km North and another of 4 km to the East what is the car's displacement? A 5#2 4 km B 7 C 5 D 4 E 3 3 km x
Slide 27 / 159 Intro to Vectors 14 If a car under goes a displacement of 7 km North and another of 3 km to the West what is the car's displacement? A B C D E 3 km x 7 km
Slide 28 / 159 Intro to Vectors 15 If a car under goes a displacement of 12 km South and another of 5 km to the East what is the car's displacement? A B C D E
Slide 29 / 159 Intro to Vectors 16 What is the magnitude of?
Slide 30 / 159 Intro to Vectors 17 What is the magnitude of?
Slide 31 / 159 Intro to Vectors 18 What is the magnitude of that connects (2,1) to (5,10)?
Slide 32 / 159 Intro to Vectors 19 What is the magnitude of that connects (-4,3) to (-5,-3)?
Slide 33 / 159 Intro to Vectors 20 The components of vector A are given as follows: The magnitude of A is closest to: A 2.9 B 6.9 C 9.3 D 18.9 E 47.5
Slide 34 / 159 Intro to Vectors 21 The components of vector A are given as follows: The magnitude of A is closest to: A 4.2 B 8.4 C 11.8 D 18.9 E 70.9
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Slide 36 / 159 Intro to Vectors A plane is traveling with an eastern component of 40 miles and a northern component of 75 miles, every hour. Draw a representation of this, include the vector of the planes actual path. How many degrees North of East is the plane traveling? This is called the vectors direction.
Slide 37 / 159 Intro to Vectors 22 What is the direction of that connects (2,1) to (5,10)?
Slide 38 / 159 Intro to Vectors 23 What is the direction of that connects (-4,3) to (-5,-3)?
Slide 39 / 159 Intro to Vectors 24 The components of vector A are given as follows: The direction of A,measured counterclockwise from the east, is closest to: A 52.7 B 55.3 C 62.3 D 297.7 E 307.3
Slide 40 / 159 Intro to Vectors 25 The components of vector A are given as follows: The direction of A,measured counterclockwise from the east, is closest to: A 37.3 B 52.7 C 139.6 D 307.3 E 322.7
Slide 41 / 159 Converting Rectangular and Polar Forms Return to Table of Contents
Slide 42 / 159 Converting Rectangular and Polar Forms A vector in standard position can be written in 2 ways: When a vector is given as (x,y) it is in rectangular form. When a vector is given (r,#) it is in polar form.
Slide 43 / 159 Converting Rectangular and Polar Forms To Convert from Rectangular to Polar (-6,8)
Slide 44 / 159 Converting Rectangular and Polar Forms 26 Vector A is in standard position and in rectangular form (2,7). What is its magnitude (ie radius)?
Slide 45 / 159 Converting Rectangular and Polar Forms 27 Vector A is in standard position and in rectangular form (2,7). What is its direction?
Slide 46 / 159 Converting Rectangular and Polar Forms 28 Vector A is in standard position and in rectangular form (-3,8). What is its magnitude (ie radius)?
Converting Rectangular and Polar Forms Slide 47 / 159 29 Vector A is in standard position and in rectangular form (-3,8). What is its direction?
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Converting Rectangular and Polar Forms Slide 49 / 159 30 Vector A is in standard position and in polar form (3,70). What is A x?
Converting Rectangular and Polar Forms Slide 50 / 159 31 Vector A is in standard position and in polar form (3,70). What is A y?
Slide 51 / 159 Converting Rectangular and Polar Forms 32 Vector A is in standard position and in polar form (3,110). What is A x?
Slide 52 / 159 Converting Rectangular and Polar Forms 33 Vector A is in standard position and in polar form (3,110) What is A y?
Slide 53 / 159 Operations with Vectors Return to Table of Contents
Slide 54 / 159 Scalar Multiples Return to Table of Contents
Slide 55 / 159 Scalar Multiples Scalar versus Vector A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated with it, such as velocity and acceleration. A vector is denoted by an arrow above the variable,
Slide 56 / 159 Scalar Multiples A car travels East at 50 mph, how far does it go in 3 hours? East at 50 mph is represented by a vector 3 hours is a scalar multiplier, which is why there are 3 vectors. The car traveled 150 miles in 3 hours.
Slide 57 / 159 Scalar Multiples 34 What is speed? A B Vector Scalar
Slide 58 / 159 Scalar Multiples 35 Which diagram represents a person walking Northeast at 4 mph for 2 hours? A B C D
Slide 59 / 159 Scalar Multiples When vectors are on the coordinate plane a scalar can be used on the ordered pair(s). joins (2, -4) to (4, 4) joins (4, -8) to (8, 8) What happened to the magnitude? What happen to the direction?
Slide 60 / 159 Scalar Multiples Given Find: and
Slide 61 / 159 Scalar Multiples 36 Given vector =(4, 5) what is A (8, 10) B (6, 7) C (16, 25) D (8, 7)
Slide 62 / 159 Scalar Multiples 37 Given vector =(4, 5) what is A (8/3, 10/3) B (2, 7/3) C (16/3, 25/3) D (8/3, 7/3)
Slide 63 / 159 Scalar Multiples 38 Given vector =(4, 5) what is A (1, 2) B (12, 15) C (-12, 15) D (-12, -15)
Slide 64 / 159 Scalar Multiples 39 Given find
Slide 65 / 159 Addition Return to Table of Contents
Slide 66 / 159 Addition Vector Addition
Slide 67 / 159 Addition Vector Addition Methods Tail to Tip Method
Slide 68 / 159 Addition Vector Addition Methods Move the vectors to represent the following operation. Draw the resultant vector.
Slide 69 / 159 Addition Vector Addition Methods Parallelogram Method Place the tails of each vector against one another. If you finish drawing the parallelogram with dashed lines and draw a diagonal line from the tails to the other end of the parallelogram to find the vector sum.
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Slide 71 / 159 Addition Given Find the resultant vector:
Slide 72 / 159 Addition 40 What is the resultant vector for if A (5, 5) B (4, 6) C (5, 6) D (4, 5)
Slide 73 / 159 Addition 41 What is if
Slide 74 / 159 Addition 42 The components of vectors and are given as follows: Solve for the magnitude of A 5 B #17 C 17 D 10 E 8
Slide 75 / 159 Subtraction Return to Table of Contents
Slide 76 / 159 Subtraction Anti- Parallel Vectors
Slide 77 / 159 Subtraction
Slide 78 / 159 Subtraction Vector Addition Method for Subtraction Draw the vectors to represent the following operation. Draw the resultant vector.
Slide 79 / 159 Subtraction Vector Addition Method for Subtraction Draw the vectors to represent the following operation. Draw the resultant vector.
Slide 80 / 159 Subtraction Vector Addition Method for Subtraction Draw the vectors to represent the following operation. Draw the resultant vector.
Slide 81 / 159 Subtraction Vector Addition Method for Subtraction Draw the vectors to represent the following operation. Draw the resultant vector.
Slide 82 / 159 Subtraction 43 A B C D
Slide 83 / 159 Subtraction 44 A B C D
Slide 84 / 159 Subtraction 45 A B C D
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Slide 86 / 159 Subtraction Given Find the resultant vector:
Slide 87 / 159 Subtraction 46 What is the resultant vector for if A (3, 1) B (-3, -1) C (5, 5) D (3, -1)
Slide 88 / 159 Subtraction 47 What is if
Slide 89 / 159 Subtraction 48 The components of vector A and B are given as follows: The magnitude of B-A, is closest to: A 10.17 B 4.92 C 2.8 D 9.7 E 25
Slide 90 / 159 Vector Equations of Lines Return to Table of Contents
Slide 91 / 159 Vector Equations of Lines Vector Equation for a Line Consider the line through R and S. There is a unique congruent vector, in standard position. R S The difference between any two points on the line is where t is a real number.
Slide 92 / 159 Vector Equations of Lines Vector Equation for a Line Example: Find the equation of the line through R(2,6) and is parallel to v. R S
Slide 93 / 159 Vector Equations of Lines Draw a graph of the line through (2, 7) and parallel to v=(1,4) Write the equation of the line. Write the equation of the line in parametric form.
Slide 94 / 159 Vector Equations of Lines Draw a graph of the line through (3, -7) and parallel to v=(-2,6) Write the equation of the line. Write the equation of the line in parametric form.
Slide 95 / 159 Vector Equations of Lines 49 Which of the following is the vector equation of the line through (-3, -4) and parallel to u=(6, 1)? A B C D (x+4, y+3) = t(6,1) (x-4, y-3) = t(6,1) (x+3, y+4) = t(6,1) (x-3, y-4) = t(6,1)
Slide 96 / 159 Vector Equations of Lines 50 Which of the following is parametric form of the equation of the line through (-3, -4) and parallel to u=(6, 1)? A C B D
Slide 97 / 159 Vector Equations of Lines 51 Which of the following is the vector equation of the line through (5, 2) and parallel to u=( -7, 1)? A B C D (x -1, y+7) = t(5,2) (x+7, y-1) = t(5,2) (x+5, y+2) = t(-7,1) (x-5, y-2) = t(-7,1)
Slide 98 / 159 Vector Equations of Lines 52 Which of the following is parametric form of the equation of the line through (5, 2) and parallel to u=( -7, 1)? A C B D
Slide 99 / 159 Vector Equations of Lines Given two points (x 1,y 1 ) and (x 2,y 2 ), the parametric equations of the line are: x = x 1 + t*(x 2 - x 1 ) y = y 1 + t*(y 2 - y 1 ) Find the parametric equation of the line through (4, 7) and (2, 8)
Slide 100 / 159 Vector Equations of Lines Find the parametric equation of the line through (3, -5) and (8, 9)
Slide 101 / 159 Vector Equations of Lines 53 Which of the following is a parametric form of the equation of the line through ( -2, 0) and (4, 7)? A C B D
Slide 102 / 159 Vector Equations of Lines 54 Which of the following is a parametric form of the equation of the line through (4, 9) and (8,3)? A C B D
Slide 103 / 159 Vector Equations of Lines Find the vector equation of:
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Slide 105 / 159 Dot Product Return to Table of Contents
Slide 106 / 159 Dot Product The dot product, also called a scalar product, returns a single numerical value between 2 vectors. Example: Given: Find:
Slide 107 / 159 Dot Product Given Find the resultant vector:
Slide 108 / 159 Dot Product 56Given, find
Slide 109 / 159 Dot Product 57Given, find
Slide 110 / 159 Dot Product 58Given, find
Slide 111 / 159 Dot Product If Vectors are perpendicular/orthogonal/normal, if their dot product is zero. and Are the vectors perpendicular? Justify your answer.
Slide 112 / 159 Dot Product If Vectors form an obtuse angle if dot product < 0 Vectors form an acute angle if dot product > 0 and What kind of angle do the vectors form? Justify your answer.
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Slide 114 / 159 Dot Product 59Are these vectors perpendicular? Yes No
Slide 115 / 159 Dot Product 60Are these vectors orthogonal? Yes No
Slide 116 / 159 Dot Product 61Are these vectors normal? Yes No
Slide 117 / 159 Dot Product 62What kind of angle is formed by the following vectors? A B C acute right obtuse
Slide 118 / 159 Dot Product 63What kind of angle is formed by the following vectors? A B C acute right obtuse
Slide 119 / 159 Dot Product 64What kind of angle is formed by the following vectors? A B C acute right obtuse
Slide 120 / 159 Angle Between Vectors Return to Table of Contents
Slide 121 / 159 Angle Between Vectors Angle Between Vectors is the dot product of the vectors is the product of the magnitudes
Slide 122 / 159 Angle Between Vectors Find the angle between and
Slide 123 / 159 Angle Between Vectors Recall that during the study of dot product that an angle was obtuse if its dot product was negative. cos -1 returns an obtuse value only if And the only way for is if
Slide 124 / 159 Angle Between Vectors 65Find the angle between and
Slide 125 / 159 Angle Between Vectors 66Find the angle between and
Slide 126 / 159 Angle Between Vectors 67Find the angle between and
Slide 127 / 159 Angle Between Vectors 68A cruise ship is being towed by 2 tug boats. The first is pulling the ship 100m east and 20m north. The second is pulling the ship 40m east and 75m north. What is the angle between the tow lines?
Slide 128 / 159 Angle Between Vectors 69A cruise ship is being towed by 2 tug boats. The first is pulling the ship 100m east and 20m north. The second is pulling the ship 40m east and 75m north. What is the angle the ship is going relative to east?
Slide 129 / 159 3-Dimensional Space Return to Table of Contents
Slide 130 / 159 3-Dimensional Space z (+) (-) (-) y (+) (-) x (+)
Slide 131 / 159 3-Dimensional Space z Graph (2,3,4) (+) (-) (-) y (+) (-) x (+)
Slide 132 / 159 3-Dimensional Space z Graph (-2, 3, -4) (+) (-) (-) y (+) (-) x (+)
Slide 133 / 159 3-Dimensional Space z (+) Graph (0,3,5) (-) (-) y (+) (-) x (+)
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Slide 135 / 159 3-Dimensional Space 70What are the coordinates of the point? A (1,3,6) B (1,6,3) C (3,1,6) 3 6 D (3,6,1) 1
Slide 136 / 159 3-Dimensional Space 71What are the coordinates of the point? A (5,0,2) B (0,2,5) C (2,0,5) 2 D (2,5,0) 5
Slide 137 / 159 3-Dimensional Space 72What is the distance between (2,3,4) and (-2,-3,-4)?
Slide 138 / 159 3-Dimensional Space 73What is the distance between (3,0,7) and (-6,-2,5)?
Slide 139 / 159 3-Dimensional Space 74What is the length of a diagonal ofa box with dimesions 4 x 6 x 7?
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Slide 144 / 159 Vectors, Lines, and Planes Return to Table of Contents
Slide 145 / 159 Vectors, Lines, & Planes Vector Operations in 3-Space Addition: Subtraction: Scalar Multiplication: Dot Product: Angle Between Vectors: Orthogonal Vectors: Dot product =0
Slide 146 / 159 Vectors, Lines, & Planes Find: What is the angle between and
Slide 147 / 159 Vectors, Lines, & Planes Lines in Space Point P = (2, 6, -3) is on line m and parallel to v = (4, -4, 2). Write the equation of m. Vector Equation Parametric
Slide 148 / 159 Vectors, Lines, & Planes Equation of a Plane 6 3x +4y +2z =12 The intersection of the plane and the xy-plane is line. So although the figure looks like a triangle it continues forever. 3 4
Vectors, Lines, & Planes Slide 149 / 159 Equation of a Plane 6 3x +4y +2z =12 The vector v=(3, 4, 2) is perpendicular to the plane. 3 4
Slide 150 / 159 Vectors, Lines, & Planes Example: Write the equation of the line perpendicular to 3x +8y - 12z =24 passing through P=(1, -5, 7).
Slide 151 / 159 Vectors, Lines, & Planes Example: Write the equation of plane passing through (1, 3, 5) and perpendicular to v = (-2, 3, 4).
Slide 152 / 159 Vectors, Lines, & Planes 79What is the x-intercept of 3x + 8y + 12z = 24?
Slide 153 / 159 Vectors, Lines, & Planes 80What is the y-intercept of 3x + 8y + 12z = 24?
Slide 154 / 159 Vectors, Lines, & Planes 81What is the z-intercept of 3x + 8y + 12z = 24?
Slide 155 / 159 Vectors, Lines, & Planes 82Which is the equation of the line perpendicular to 3x +8y +12z = 24 and passing through (2, -4, 6)? A (x+2, y-4, z+6)=t(3, 8, 12) B (x-2, y+4, z-6)=t(3, 8, 12) C (x+2, y+4, z+6)=t(3, 8, 12) D (x-2, y-4, z-6)=t(3, 8, 12)
Slide 156 / 159 Vectors, Lines, & Planes 83Which is the equation of the line perpendicular to x -2y +7z = 14 and passing through (-1, 4, -8)? A (x+1, y-4, z+8)=t(1, -2, 7) B (x+1, y-4, z+8)=t(-1, 2, -7) C (x-1, y+4, z-8)=t(1, -2, 7) D (x-1, y+4, z-8)=t(-1, 2, -7)
Slide 157 / 159 Vectors, Lines, & Planes Cross Product 2 non-parallel (or anti-parallel) vectors lie in one plane. The Cross Product finds a vector perpendicular to that plane.
Slide 158 / 159 Vectors, Lines, & Planes Cross Product Find a vector perpendicular to plane formed by
Slide 159 / 159 Vectors, Lines, & Planes 84Which of the following is perpendicular to the plane formed by a= (3, -1, 2) and b= (4, -5, 6) A B C D