CONTENTS. INTRODUCTION MEQ curriculum objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4

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1 CONTENTS INTRODUCTION MEQ crriclm objectives for vectors (8% of year). page 2 What is a vector? What is a scalar? page 3, 4 VECTOR CONCEPTS FROM GEOMETRIC AND ALGEBRAIC PERSPECTIVES page 1 Representation of a vector, and vector notation 5 2 Magnitde (norm) of a vector 6 3 Special Vectors (nit vector, zero vector) 7 4 Vector Relationships (opposite vectors, eqivalent 8 9 vectors, collinear vectors, orthogonal vectors) 5 Vector addition 10 6 Vector sbtraction 11 7 Mltiplication of a vector by a scalar 12 8 Scalar or Dot Prodct 13 9 Properties of Operations on vectors Basis of a Vector Space 15 APPLICATIONS Propositions which stdents have to verify or prove. page 16 APPENDIX Strategies for proofs from Mathématiqes 2000 page 17 Website list for resorces to nderstand vectors page 18 Stdent s previos backgrond for vector stdy. page MEQ 1999 exam qestions on vector concepts page Geometer s Sketchpad activities for vector propositions Cabri Geometry II activities for vector propositions Math 536 Vectors page 1

2 MEQ VECTOR OBJECTIVES (8% of year) (refer to pages and in crriclm gide) Stdents will be introdced to two aspects of the theory of vectors: vectors represented by arrows in a plane or as ordered pairs of real nmbers in the Cartesian plane. They will stdy vector addition, the mltiplication of a vector by a scalar and the scalar prodct of two vectors. Stdents will also be able to demonstrate propositions sing vectors. Finally, the stdents will be given problems related to what they have jst learned abot vectors, as well as real-life problems that involve applying these principles. (etc.) Intermediate Objectives To perform the following operations on vectors: addition, mltiplication by a scalar and the scalar mltiplication of two vectors. To determine the properties of operations on vectors. To define the concept of the basis of a vector space. To prove propositions related to vectors. To prove propositions sing vectors. To jstify statements sed in solving a problem. A list of propositions can be fond on pages of the crriclm gide. These are repeated later in this docment. Please note that throghot this docment, there are sample exercises sggested from the stdent text, Mathematical Reflections, Book 2, Reflection 6 (CEC). These exercises are meant to be samples only. Please refer to the MAPCO 536 Reflections gide for this chapter for other appropriate exercises. Math 536 Vectors page 2

3 WHAT IS A VECTOR? A directed line segment. A line segment which has magnitde and direction. A vector can represent displacement, velocity, acceleration and force. A vector can be represented by a directed line segment (arrow). If a vector is moved by parallel displacement, its direction and magnitde remain nchanged. These arrows below all represent the same vector. Exercise: page 123 part a. Please note that if yo are referring to one of the French textbooks for vector concepts, yo will notice that there are three criteria for vectors: longer (magnitde), sens (direction) and direction (orientation). Sens refers to which way the head of the vector is pointing, while direction refers to the slope of the vector. Since or English text refers only to magnitde and direction, this docment follows this scheme. Math 536 Vectors page 3

4 WHAT IS A SCALAR? A real nmber that can be represented on a scale or nmber line. i.e. a scalar has magnitde, bt no direction. EXAMPLES: VECTORS SCALARS 50 km, N 30 degrees E 34 degrees Celsis 89 kg 6 km 100 km/h 188 cm 3 7 hors Exercise: page 129 # 1 Also from MAPCO binder Jne 1999: For each of the following, determine which sitations involve scalars and which involve vectors: a. plling a rope at an angle of 45 degrees with all my might b. the otside thermometer reading of 23 o C c. pshing a water wheel with a force of 15 N d. the space station moving from West to East at km/h e. a polygon of perimeter 3 cm which represents the area of a field of 30 m 2 f. a line segment, AB, 5 cm long, at an angle of 45 o to the x-axis, representing a boat sailing towards the Northeast at 25 knots g. a cargo container being loaded onto a ship h. the amont of gas in a tank which is almost empty i. slowly backing ot of a parking spot at the mall j. having enogh water in the fish tank at home Answers: Scalars b e h j Vectors a c d f g i Math 536 Vectors page 4

5 GEOMETRIC and ALGEBRAIC 1. REPRESENTATION OF A VECTOR AND VECTOR NOTATION Vectors do not have position. The displacement from one point to another doesn t tell s where we started or ended, only how we moved. GEOMETRIC APPROACH Every vector has a tail (initial point) and a head (terminal point). tail head Two points A and B, taken in this order, represent a vector AB. A vector AB is represented by an arrow going from A to B. AB A DC C D Vector directions can also be identified by reference to compass headings. N 30 o E B ALGEBRAIC APPROACH Vectors are represented by ordered pairs in a Cartesian plane. In the ordered pair (a, b) the x coordinate represents the horizontal component of the vector, and the y coordinate represents the vertical component of the vector. = (a, b) (a, b) When a vector s tail does not originate at (0, 0) the horizontal and vertical components mst be calclated. (x 1, y 1 ) (x 2, y 2 ) 129, qestion 2 and page 130 qestion 3. = (x 2 x 1, y 2 y 1 ) = (a, b) 130, # 7, #14 Math 536 Vectors page 5

6 2. MAGNITUDE (NORM) OF A VECTOR The magnitde or norm of a vector is the length of the vector. The magnitde of vector is written as. GEOMETRIC APPROACH The length of a vector can be measred if the drawing is to scale. ALGEBRAIC APPROACH The length of a vector can be calclated by Pythagorean relationships. A 25 km (a, b) AB = 25 km B 125, qestion f) and page 131 qestion 17. = 2 a + b 2 The length can also be calclated sing the Pythagorean Theorem, the Sine Law, or the Cosine Law, depending on the information given. (x 1, y 1 ) (x 2, y 2 ) 48 cm = (x x 1) (y 2 - y 1) 36 cm = , qestions 18 a, b 131 qestions 11 and 12. Math 536 Vectors page 6

7 3. SPECIAL VECTORS: UNIT VECTOR, ZERO VECTOR Definitions: A nit vector has a magnitde of 1 nit. = 1 A zero vector has no length, and can be = 0 considered to have any direction. It has the same beginning and ending point. GEOMETRIC APPROACH Unit vector: 1 m ALGEBRAIC APPROACH Unit vector: (-2, 2) (-2, 1) (1,0) Zero vector: AA = 0 A nit vector with its tail at the origin will have its head on the standard nit circle. Zero vector: 0 = (0, 0) Math 536 Vectors page 7

8 4. VECTOR RELATIONSHIPS a) OPPOSITE VECTORS: Opposite vectors have the same magnitde bt opposite directions They are parallel. Every vector has an opposite vector. GEOMETRIC APPROACH Opposite vector: ALGEBRAIC APPROACH Opposite vector: AB AB = BA = (a, b) = ( a, b) A 60 m A B 60 m B 134 qestion qestion 9 b) EQUIVALENT VECTORS (also called EQUAL Vectors) When arrow AB and arrow CD represent the same vector, we write: AB CD GEOMETRIC APPROACH Vectors which are eqivalent to each other have the same direction and the same magnitde. B A 5 km ALGEBRAIC APPROACH Vectors which are eqivalent to each other have the same components. Given vectors and v, sch that AB CD C 5 km (qadrilateral ABDC will be a parallelogram) 126, qestion h. D = (a 1, b 1 ) and v = (a 2, b 2 ) v a 1 = a 2 and b 1 = b 2 We can also write (a 1, b 1 ) (a 2, b 2 ) Vectors can be eqivalent, or eqal. Arrows which represent the same vector are eqipollent. Math 536 Vectors page 8

9 c) COLLINEAR VECTORS: When arrows representing two vectors are parallel, the vectors are collinear, or linearly dependent. The length of one vector will be a mltiple of the length of the other vector. GEOMETRIC APPROACH and v are collinear k ε R sch that v = k. v ALGEBRAIC APPROACH = (a, b) and v = (c, d) are collinear ad bc = 0. For example: v = 2 = (3, 5) and v = (6, 10) ad bc = 3(10) 5(6) = 0 v = 2.5 d) ORTHOGONAL VECTORS: Two vectors are orthogonal if and only if the arrows representing them are perpendiclar to each other. Exercises to review terms for these 4 relationships: page 134, qestion 4 and page 136, qestion 10. Math 536 Vectors page 9

10 5. VECTOR ADDITION Vector addition may represent the sm of forces, displacements, velocities or accelerations. When two vectors are added, the vector sm is called the resltant vector. GEOMETRIC APPROACH Michel Chasles ( ) discovered a relation that exists for vector sms. This is sefl when the vectors can be considered seqentially. The resltant vector is the vector from the initial starting point to the final ending point. C AB + BC = AC A If two vectors are considered from the same initial point, the parallelogram law of vector addition can be sed. Consider the two vectors as adjacent sides of a parallelogram, draw the other two sides, and the diagonal from the vertex of the two tails. This diagonal represents the sm of the two vectors. B ALGEBRAIC APPROACH Given = (a, b) and v = (c, d) A + v = (a + c, b + d) C B v + v D Example: = AB and v = CD where A = ( 4, 6) and B = (1, 5) C = (1, 7) and D = (8, 12) + v v = (1 ( 4), 5 6) = (5, 1) v = (8 1, 12 7) = (7, 5) + v = (5 + 7, 1 + 5) = (12, 4) In either case, calclate the magnitde of the resltant sing the cosine law. (Remember that the sm of the angles in a parallelogram is 360 o.) 141 c, 148, # 14 and , qestion 8 (draw on grid), 147, qestion 12; pages Math 536 Vectors page 10

11 6. VECTOR SUBTRACTION GEOMETRIC APPROACH When two vectors are considered seqentially, the difference of the vectors is fond by adding the opposite vector. ALGEBRAIC APPROACH Given = (a, b) and v = (c, d) v = (a c, b d) AB BC = AB + ( BC) or, AB BC = AB + CB C A B AB - BC BC When two vectors are considered from the same initial point, sbtraction of vectors can be viewed as the diagonal connecting the two heads in a parallelogram with the two initial vectors as adjacent sides. v v (Note that the head of the difference vector points towards the head of the first vector.) 141, k; 142 l, m; 143, n, o; 144, s; 146, 6 and 8; 147, 9 Math 536 Vectors page 11

12 7. MULTIPLICATION OF A VECTOR BY A SCALAR The mltiplication of a vector by a positive scalar reslts in a vector having the same direction as the initial vector, with a length in proportion to the scalar (scale factor). If the scalar is negative, the direction of the reslting vector will be opposite to that of the original vector. Example: 3 = + + GEOMETRIC APPROACH ALGEBRAIC APPROACH Given scalar k and vector = (a, b) k k > 1 k = k (a, b) = (ka, kb) k 0 < k < 1 k k k < 0 154, qestions 6 and , qestion 12 a. 153, qestion 1 a, b and 2 a 155, qestion 11 a and b. Note that k is a vector, and k is a scalar (magnitde only). Note also that k = k x Math 536 Vectors page 12

13 8. VECTOR SCALAR PRODUCT (or DOT PRODUCT) The scalar prodct of two vectors is a scalar. The scalar prodct is written as: AB CD or v In physics, the scalar prodct gives the Work Done when a force is applied to an object and it displaces it a certain distance in a certain direction. This is represented by the formla W = Fd. (1 Newton metre = 1 jole) GEOMETRIC APPROACH v = v cos θ where θ is the angle between ALGEBRAIC APPROACH AB CD = (a, b) (c, d) = ac + bd vectors and v. applied force F a θ displacement d actal force F h = F a cos θ 160, qestion 4 a and b 161, qestion , qestion 1 160, qestion 6 When two vectors are perpendiclar (orthogonal), the angle between them is 90 degrees, and the cosine of that angle is 0. Ths: v v = 0 When θ = 0 (vectors are collinear), cos θ = 1, (work is done efficiently). When θ > 90 o, cos θ < 0 and therefore v < 0 (like taking a dog for a walk!) Properties of scalar (dot) prodcts: Commtative: v = v Associative: (k ) v = k ( v ) Distribtive: w ( + v ) = (w ) + (w v) Math 536 Vectors page 13

14 9. PROPERTIES OF OPERATIONS ON VECTORS (see Topic 4) Commtative properties of addition and mltiplication: AB + BC = BC + AB + v = v + (k m) = (m k) Associative properties of addition and mltiplication: (AB + BC) + CD = AB + (BC + CD) ( + v ) + w = + ( v + w ) k ( m ) = ( k m ) Distribtive property of mltiplication over addition or sbtraction: k (AB ± CD) = k AB ± k CD k ( ± v ) = k ± k v (k ± m) = k ± m Identity Element for addition the zero vector: AB + 0 = AB (x, y) + (0, 0) = (x, y) Identity Element for mltiplication the nit vector. Additive Inverse the opposite vector: AB + ( AB) = 0 + ( ) = (a, b) + (a, b) = (a, b) + ( a, b) = (0, 0) Exercises pages 166 and 167. Math 536 Vectors page 14

15 10. BASIS OF A VECTOR SPACE This topic relates to collinear vectors where v = k. It also relates to the addition of vectors, and their resltant. Any vector can be written as a linear combination of two basis vectors, as long as the basis vectors aren t parallel to each other (are linearly independent). GEOMETRIC APPROACH b b a = 3 a + 2 b Vectors a and b represent basis vectors, and is a vector represented by a linear combination of that vector base. 176, qestion 3, 5 a a b a ALGEBRAIC APPROACH The Cartesian plane has a vector basis formed by two nit vectors, i and j, which are parallel to the x and y axes respectively, and are orthogonal to each other. 1 j i 1 = 4 i + 2 j i = (1, 0) and j = (0, 1) so = 4 (1, 0) + 2 (0, 1) (4, 2) All vectors can be expressed as a linear combination of i and j. 174, qestion f The Cartesian plane is an example of an orthonormal coordinate system with basis vectors which are nit vectors, orthogonal to each other. Math 536 Vectors page 15

16 PROPOSITIONS: Let, v and w be vectors in the plane and r and s, scalars. Verify the following properties: (r = 0) (r = 0 or = 0) 2. if and v are non-collinear vectors then (r = sv) (r = s = 0) 3. (w is collinear with ) ( r ε R: w = r) 4. ( and v are non-collinear) ( w, r ε R, s ε R: w = r + sv) 5. ( v ) ( v = 0) Using vectors, prove the following propositions: 188 (top) (top) 193 # # (bottom) 194 # # The diagonals of a parallelogram bisect each other. In addition, a qadrilateral whose diagonals bisect each other is a parallelogram. 7. The line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of the third side. 8. The midpoints of the sides of any qadrilateral are the vertices of a parallelogram. 9. If O is a point in the plane, and if P, Q and R are the midpoints of the sides of triangle ABC, then OP + OQ + OR = OA + OB + OC 10. In any parallelogram, if a vertex is joined to the midpoints of the nonadjacent sides, then the opposite diagonal is ct into three eqal segments. 11. The medians of a triangle are concrrent in a point that is two-thirds the distance from each vertex to the midpoint of the opposite side. 12. An angle inscribed in a semicircle is a right angle. 13. In any triangle, the sqare of the length of a side is eqal to the sm of the sqares of the length of the other two sides, mins twice the prodct of the lengths of the other two sides and the cosine of the contained angle. (The Cosine Law) 14. The diagonals of a rhombs are perpendiclar. 15. If the midpoints of the sides of an eqilateral triangle are joined, an eqilateral triangle is formed, each side of which measres half that of the original triangle. Math 536 Vectors page 16

17 (translated from Mathématiqes 2000) Helps in proving propositions sing vectors: 1. To prove that qadrilateral ABCD is a parallelogram, yo can prove that: AB = DC or that AD = BC. 2. To prove that line segments d and d are parallel, yo can prove that: the vectors of d and d are collinear; or that the normalized vectors of d and d are collinear 3. To prove that the line segments d and d are perpendiclar, yo can prove that: the vectors of d and d are orthogonal; or that the normalized vectors of d and d are orthogonal; or that the scalar prodct of the vectors of d and d is zero. 4. To prove that M is the middle of segment AB yo can prove that: the vectors AM and MB are eqal; or that the vectors MA and MB are opposite. In all vector proofs, remember Chasles Relation to simplify vector sms. See also page 178 in Reflections for additional strategies. Math 536 Vectors page 17

18 WEBSITES TO VISIT (ON VECTORS) cs/vectors/index.html Math 536 Vectors page 18

19 WHAT PRIOR CONCEPTS HAVE STUDENTS ENCOUNTERED? MATH 116 Translations in the plane Stdents are given a polygon as an object, and a translation arrow which shows direction and distance. They mst translate the polygon vertex by vertex, by drawing lines parallel to the translation arrow, and measring off the length of the arrow. This gives them the image of the polygon nder the given translation. MATH 116 Nmber Line for Integer Operations Some textbooks explain integer addition and sbtraction with the se of directed line segments ( 8) Math 536 Vectors page 19

20 MATH 216 TRANSFORMATIONS IN THE COORDINATE PLANE Stdents are given a polygon on a coordinate plane, and a mapping rle for translations and dilatations. They then operate on the coordinates of the vertices of the polygon according to the mapping rle in order to constrct the image of the polygon nder the transformation. t (-2,4) : (x,y) a (x 2, y + 4) Example: coordinates of vertex A are (5, 2) Coordinates of vertex A nder the translation are (5 2, 2 + 4) = (3, 6) h (0,3) : (x,y) a (3x, 3y) Example: coordinates of vertex P are (2, 3) Coordinates of vertex P nder the dilatation are (3 x 2, 3 x 3) = (6, 9) MATH 314 COMPOSITIONS OF TRANSFORMATIONS Stdents are asked to carry ot two consective translations of an object, then to identify a single translation that wold have the same effect as the composite of the two translations. They are also asked for the translation that wold take the image back to the position of the object, or the inverse translation. Given t 1 : (x,y) a (x + 2, x 5) and t 2 : (x,y) a (x, y + 2) Constrct: t 2 N t 1 Example: coordinates of vertex M are ( 3, 1) Under t 1 the coordinates of M are ( 1, 4) Under t 2 the coordinates of M are ( 1, 2) The composite translation t 2 N t 1 is (x, y) a (x + 2, y 3) The composite inverse transformation (t 2 N t 1 ) 1 is (x, y) a (x 2, y + 3) Math 536 Vectors page 20

21 MEQ 1999 Exam Qestions On Vector Concepts Q1. Q2. Math 536 Vectors page 21

22 Q3. Math 536 Vectors page 22