UNIT 1 INTRODUCTION TO VECTORS Lesson TOPIC Suggested Work Sept. 5 1.0 Review of Pre-requisite Skills Pg. 273 # 1 9 OR WS 1.0 Fill in Info sheet and get permission sheet signed. Bring in $3 for lesson shells & $7 if you need a calculator Sept. 6 1.1 (1) 6.1 Introduction to Vectors Pg. 279 # 1 6, 8, 9, 10 OR WS 1.1 Sept. 7 1.2 (2) 6.2 Vector Addition Pg. 290 # 3, 4, 6, 7, 11 14 OR WS 1.2 Sept. 8 1.3 (3) 6.3 Multiplication of a Vector by a Scalar Pg. 298 # 4, 5, 7a, 11, 13, 15, 17, 19 OR WS 1.3 Sept. 11 1.4 (4) 6.4 Properties of Vectors Pg. 306 # 3, 5, 7, 8ac, 9, 11 OR WS 1.4 Sept. 12 1.5 (5) 6.5 Vectors in R 2 and R 3 Pg. 316 # 3, 5, 6, 7, 9, 11, 13, 14, 15 OR WS 1.5 Sept. 13 1.6 (6) 6.6 Operations with Algebraic Vectors in R 2 Pg. 324 # 1, 3, 4, (5, 6)b, 8bd, 10, (13, 15)a Sept. 14 1.7 (7) 6.7 Vectors in R 3 Pg. 332 # 1, 2, 4, (5, 6)ac, 8, 9, 11, 12 Sept. 15 1.8 (8) 6.8 Linear Combinations and Spanning Sets Pg. 340 # 1, 3, 5, 7b, 9bd, 11, 12, 13b, 14, 15 Sept. 18 1.9 (9) Review for Unit 1 Test Pg. # 344 # 2a, 3 6, 7ab, 8, 11a, 12, 13, 15, 16, 18, 19, 21 Sept. 20 1.10 (10) TEST- UNIT 1
MCV 4U Lesson 1.1 Introduction to Vectors A vector is a mathematical quantity having both magnitude (size) and direction. Velocity is a vector - 60 km/h North is a vector quantity as is 40 Newtons down. A scalar is a mathematical quantity has magnitude or size only. NO DIRECTION. Speed is a scalar - 60 km/h is a scalar quantity as is 40 Newtons. A vector is represented as directed line segments with an arrow at one end to indicate the direction. The length of the line segment represents the magnitude of the vector. ALWAYS draw vector diagrams to scale. Draw angles as accurately as you can. The better the diagram, the more help it will be in solving the problem. If asked for a vector diagram DO NOT FORGET THE ARROWHEADS. (no arrows = no vector = no marks) When a single letter is used to denote a vector it is written in bold type a or it is written with an arrow over the a. It is faster to make the arrow if it only has a one sided point a ; it may also be written with a line over the a. I will use arrows with a single or double headed point. You may use whichever you wish except a. Sometimes we denote a vector using two capital letters when the vector extends from one point to another. In this case the first letter is always the starting or initial point of the vector and the second letter is the end or terminal point. B terminal point (tip) AB A Initial point (tail) The magnitude of a vector is denoted by placing absolute value signs around the vector. The magnitude of a is a. The magnitude of AB is AB. Vectors are equal iff their magnitudes and directions are the same. Opposite vectors have the same magnitude but opposite directions. The opposite of AB is AB or BA. When vectors have the same or opposite direction they are drawn parallel. In the diagram below, u v. u v Ex. If u 2 in the diagram below, draw 2 u u
Thus, multiplication of a vector by a scalar number results in a new vector parallel to the original but with a different magnitude. In general, two vectors u and v are parallel iff u = k v. AB BC AC If we go from point A to point B and then from point B to point C, the result is the same as going from point A to point C. B C A The angle between two vectors u and v is measured when the vectors are drawn tail to tail. The angle between them is always less than or equal to 180. u v Any vector v has a unit vector vˆ that has the same direction as v but a magnitude of 1. Recall that the magnitude of a vector is denoted by v. Therefore 1 v vˆ and v vˆ = v v A vector which has a magnitude of zero and thus a direction which is undefined (a point) is called the zero vector and is denoted by 0.
Ex. a) Given the rectangle ABCD, state: A E B 3 G H D F 8 C (i) 2 equal vectors (ii) 2 parallel vectors with different magnitudes (iii) AB as a scalar multiple of BE (iv) the angle between GC and EB b) Express AB in terms of: (i) AE (ii) FD Pg. 279 # 1 6, 8, 9, 10
MCV 4U Lesson 1.2 Vector Addition One of the primary tasks when working with vectors is finding the effect of two vectors or the sum of two vectors. The sum of two vectors has the same effect as the individual vectors. If the displacement vectors a and b move us two steps east and three steps south respectively, then the result of these two vectors will move us 2 2 3 2 13 steps in the south of east where tan = 3 2. a b The sum of two vectors is called the resultant. From the diagram we can see that if we join the two vectors together by placing them tail to tip, then the resultant is the vector that starts at the first tail and extend to the last tip. The lengths and angles involved can be calculated using trigonometry. Triangle Law of Vector Addition Place the two vectors tip to tail and the resultant extends from the first tail to the last tip. a b a + b Parallelogram Law of Addition If two vectors are tail to tail we can find the resultant by using the vectors as two adjacent sides of a parallelogram. Draw the remaining two sides of the parallelogram and the resultant extends from the common tail point to the opposite corner of the parallelogram. a a + b b
Ex. 1 For the vectors below, find: a) u + v b) u v v u Ex. 2 Consider parallelogram EFGH with diagonals EG and FH that intersect at J. a) Express each vector as the sum of two other vectors in two ways. (i) HF (ii) FH (iii) GJ b) Express each vector as the difference of two other vectors in two ways. (i) HF (ii) FH (iii) GJ Ex. 3 In an orienteering race, you walk 100 m due east and the walk N70 E for 60 m. How far are you from your original position and in what direction?
Ex. 4 Find the resultant of the following parallel vectors. a) 5 km/h E 7 km/h E b) 30 km/h E 65 km/h W c) 50 N SW 20 N NE Pg. 290 # 3, 4, 6, 7, 11 14
MCV 4U Lesson 1.3 Multiplication of a Vector by a Scalar When you multiply a vector by a scalar, the magnitude of the vector is multiplied by the scalar and the vectors are parallel. If the scalar is positive, the direction remains unchanged. If the scalar is negative, the direction becomes opposite. ie: For the vector k a, where k is a scalar and a is a nonzero vector with magnitude a : If k 0, then k a has the same direction as a and has a magnitude of k a. If k 0, then k a has the opposite direction as a and has a magnitude of k a. Two vectors a and b are collinear iff it is possible to find a non-zero scalar k such that a kb. 1 a a is a unit vector,( vector with a length of 1) in the same direction as a. 1 a a is a unit vector,( vector with a length of 1) in the opposite direction as a. Linear combinations of vectors can be formed by adding scalar multiples of two or more vectors. ie: u 2a 3b Ex. 1 a) Which of the vectors below are scalar multiples of vector v? Explain. b) Find the value of the scalar k for each vector in (a). b d v f a
Ex. 2 Consider vector u with magnitude u = 100 km/h in a direction of N40 E. Draw a vector to represent each scalar multiple and describe the resulting vector. a) 3 u b) 0.5 u c) 2 u u Ex. 3 In trapezoid ABCD, BC AD and AD = 3BC. Let AB u and BC v. Express AD, BD, and CD as linear combinations of u and v. v u B C A D
Ex. 4 a and b are unit vectors that have an angle of 20 between them. a) Find the value of 3a 2b. b) Find the direction of 3a 2b. c) Find the unit vector in the same direction as 3a 2b. Pg. 298 # 4, 5, 7a, 11, 13, 15, 17, 19
MCV 4U Lesson 1.4 Vector Properties Properties of Vector Addition Commutative Law: Associative Law: a + b = b + a a + ( b + c ) = ( a + b ) + c Properties of Scalar Multiplication Associative Law: (mn) a = m(n a ) Distributive Law: m( a + b ) = m a + m b (m + n) a = m a + n a Properties of the Zero Vector a + 0 = a Each vector a has a negative, a, such that a + ( a ) = 0 Triangle Inequality For vectors a and b, a + b a b Since vectors a and b and their sum a + b form the sides of a triangle, the lengths of the sides of the triangle are the magnitudes of the vectors. From the figure 1 below, the side a + b must be less than the sum of the other two sides, a b, otherwise there is no triangle. Therefore a + b a b. When a and b have the same direction, then we have a case where a + b a b, as seen in figure 2. a + b a figure 1 b a a + b figure 2 b Ex. 1 Simplify. 2(u 2v 4w ) 3(2u v 3w )
Ex. 2 If a 2i 3 j k and b i 3k and c 3i 2 j k, find each of the following in terms of i, j, and k. a) a b b) 2a 3b c Pg. 306 # 3, 5, 7, 8ac, 9, 11
MCV 4U Lesson 1.5 Vectors in Two Space (R 2 ) and Three Space(R 3 ) By placing vectors on the Cartesian Plane, we can use algebraic methods in our study of vectors. 2 Dimensions (R 2 ) The ordered pair (a, b) is refered to as an algebraic vector. ie: OP = (a, b) The values of a and b are called the x- and y-components of the vector. Its tail is located at (0, 0) and its head is located at (a, b). N.B. (a, b) can represent a point with coordinates a and b, or it can represent a vector with components a and b. You should be able to tell by the context of a problem which is being meant. If you see an equal sign beside the brackets this would indicate a vector. ie: A(3, 4) is a point, while a = (3, 4) is a vector. 3 Dimensions (R 3 ) (a.k.a. 3 space or space) Similarly, any vector u in space can be written as an ordered triple where: u = OP = (a, b, c) where a, b, and c are its x-, y-, and z-components Its tail is located at (0, 0) and its head is located at (a, b, c). To plot a vector in space, you move a units along the x-axis, b units parallel to the y-axis and then c units parallel to the z-axis. Drawing a rectangular box (prism) is sometimes helpful with this In R 3, the three mutually perpendicular axes form a right handed system. Right Handed System In R 2 or R 3 the location of every point is unique. As a result,every vector drawn with its tail at the origin and its head at a point is also unique. These types of vec tors are known as position vectors.
Ex. 2 Locate the point P, and sketch the position of vector OP in three dimensions. a) (2, 0, 0) b) (0, 0, 3) c) (1, 2, 0) d) (2, 3, 4) e) (-3, 4, -2) f) (4, -5, 1) g) (0, -3, -6) h) (-4, -5, -2) z y x Ex. 2 What vector is represented in each of the following diagrams? Pg. 316 # 3, 5, 6, 7, 9, 11, 13, 14, 15
MCV 4U Lesson 1.6 Operations with Algebraic Vectors in Two Space (R 2 ) Unit Vectors in Two Dimensions Define vectors î = (1, 0) and ĵ = (0, 1). The vectors of length 1 that point in the direction of the positive x-axis and positive y-axis respectively. Vector u = (a, b) can also be represented as u = ai bj. For example, the vector OA (2, 5) can be written as OA 2i 5 j We can also use the unit vectors i and j to define a vector in the plane where i = (1, 0) a unit vector along the x-axis and j = (0, 1) a unit vector along the y-axis. y 1 (i, j ) j i 1 x any vector u can also be written in the form u = a i + b j Any vector u in a plane can be written as an ordered pair (a, b), where its magnitude u and direction are given by the equations : u = a 2 b 2 b and tan = a with being measured counter clockwise from the positive x axis to the vector. Vector Addition and Subtraction in Component Form: If u = (u 1, u 2 ) and v = (v 1, v 2 ), then u + v = (u 1 + v 1, u 2 + v 2 ) and similarly u v = (u 1 v 1, u 2 v 2 ). Scalar Multiplication in Component Form: If u = (u 1, u 2, u 3 ) and k R, then k u = (k u 1, k u 2 ) Remember: Two vectors are parallel or collinear iff one vector is a scalar multiple of the other. ie: If u kv for some scalar k, then u 1 v 1 u 2 v 2 k Components of a Vector Between Two Points: Given: P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) then P 1 P 2 (x 2 x 1, y 2 y 1 ) Proof: P 1 P 1 P 2 P 1 O OP 2 P 2 OP 2 OP 1 x, y ) ( x, ( 2 2 1 y1 (x 2 x 1, y 2 y 1 ) ) O
Ex. 1 Given parallelogram ABCD with vertices A(1, 4), B(2, 3), C( 1, 4), find the coordinates of D. Ex. 2 Using vectors show that the points P(3, 7), Q(4, 8), and R(6, 10) are collinear.
Ex. 3 Given a 2i 3 j and b i 3 j determine: a) 2a 3b b) 2(3a b ) 2(a 2b ) c) 2a 3b Pg. 324 # 1, 3, 4, (5, 6)b, 8bd, 10, (13, 15)a
MCV 4U Lesson 1.7 Operations with Algebraic Vectors in Three Space (R 3 ) All vectors in space can be expressed in terms of the unit vectors i (1,0,0) and j (0,1,0) and k (0,0,1). These vectors are called basis vectors. The representation of a vector in terms of the basis vectors is unique. ie: If v a i + b j + c k, then there is only one set of values for a, b, and c. Thus, the expression of a vector in terms of a set of basis vectors is unique. If u = (a, b, c), then a, b, and c are the Cartesian components of u. 2 2 2 Its magnitude is given by u a b c Two vectors are equal iff their respective Cartesian coordinates are equal. Vector Addition and Subtraction in Component Form: If u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ), then u + v = (u 1 + v 1, u 2 + v 2, u 3 + v 3 ) and similarly u v = (u 1 v 1, u 2 v 2, u 3 v 3 ). Scalar Multiplication in Component Form: If u = (u 1, u 2, u 3 ) and k R, then k u = (k u 1, k u 2, k u 3 ) Remember: Two vectors are parallel or collinear iff one vector is a scalar multiple of the other. ie: If u kv for some scalar k, then u 1 v 1 u 2 v 2 u 3 v 3 k Components of a Vector Between Two Points: Given: P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) then P 1 P 2 (x 2 x 1, y 2 y 1, z 2 z 1 ) Proof: P 1 P 1 P 2 P 1 O OP 2 P 2 OP 2 OP 1 (x 2, y 2, z 2 ) (x 1, y 1, z 1 ) (x 2 x 1, y 2 y 1, z 2 z 1 ) The vector AB between two points with its tail at A( x 1, y 1,z 1 )A and head at B(x 2,y 2,z 2 ) is detemined AB OB OA (x 2 x 1, y 2 y 1, z 2 z 1 ).
Ex. 1 Given parallelogram ABCD with vertices A(1, 4, 6), B(2, 3, 7), C( 1, 4, 5), find the coordinates of D. Ex. 2 Given x i 3 j 2k and y 2i 3k, determine the following. a) 2x 4y b) 2x 4y Pg. 332 # 1, 2, 4, (5, 6)ac, 8, 9, 11, 12
MCV 4U Lesson 1.8 Linear Combinations & Spanning Sets Recall: Given a = (-1, 2) and b = (1, 4), find 2a 3b. The vector we created is a vector on the xy-plane and is the diagonal of the parallelogram formed by 2a and 3b. Linear Combinations of Vectors: Given noncollinear vectors u and v, a linear combination of these vectors is au + bv, where a and b are scalars. Spanning Sets: Any collinear Vectors span R 1. Any non-collinear, nonzero vectors span R 2. i.e. if you can write the vectors as a linear combination of each other, they create a spanning set in R 2. Since every vector is R 2 can be written as a linear combination of the vectors i a spanning set in R 2. and j, they form
Ex. 1 Show that = (4, 23) can be written as a linear combination of the vectors ( 1,4) and (2,5). Ex. 2 Do (3,4) and (9,12) span R 2?
This concept can be extended in R 3., and form a basis for R 3. Therefore, every vector in R 3 can be written as a linear combination of these three vectors. Spanning sets in R 3 : Any pair of nonzero, noncollinear vectors span a plane in 3-space. i.e. if you can write them as a linear combination, then they span a plane in R 3. Ex. 3 Does vector ( 9, 4,1) lie in the same plane as ( 1, 2,1) and (3, 1,1)? ie: Do they span a plane in R 3? In general, when we are trying to determine whether a vector lies in the plane determined by two other nonzero, noncollinear vectors, it is sufficient to solve any pair of equations and look for consistency in the third equation. If the result is consistent, the vector lies in the plane and the vectors span the plane, and if not, the vector does not lie in the plane. Ex. 4 Show that the vectors ( 1, 2, 3), ( 4,1, 2) and ( 14, 1,16) do not lie on the same plane. Pg. 340 #1,3, 5, 7b, 9bd, 11, 12, 13b, 14, 15