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Chapter 46 Vectors ch46 Vectors by Chtan FYKulai 1 A VECTOR? □ Describes the motion of an object □ A Vector comprises □ Direction □ Magnitude Size □ We will consider □ Column Vectors □ General Vectors □ Vector Geometry ch46 Vectors by Chtan FYKulai 2 Column Vectors NOTE! Vector a Label is in BOLD. 2 up a When handwritten, draw a wavy line under the label a i.e. ~ 4 RIGHT 4 2 COLUMN Vector ch46 Vectors by Chtan FYKulai 3 Column Vectors Vector b 2 up b 3 LEFT 3 2 COLUMN Vector? ch46 Vectors by Chtan FYKulai 4 Column Vectors Vector u 2 down n 4 LEFT 4 2 COLUMN Vector? ch46 Vectors by Chtan FYKulai 5 Describe these vectors 4 1 a 1 3 b c 2 3 d ch46 Vectors by Chtan FYKulai 4 3 6 Alternative labelling B D EF E AB F CD G C A GH H ch46 Vectors by Chtan FYKulai 7 General Vectors A Vector has BOTH a Length & a Direction All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k k k k k k can be in any position ch46 Vectors by Chtan FYKulai 8 General Vectors Line CD is Parallel to AB B A CD is TWICE length of AB k D 2k Line EF is Parallel to AB E C -k EF is equal in length to AB EF is opposite direction to AB F ch46 Vectors by Chtan FYKulai 9 Write these Vectors in terms of k B k D 2k ½k 1½k F G E A C -2k H ch46 Vectors by Chtan FYKulai 10 Combining Column Vectors 2 k 1 AB k B AB 3k D CD 2k 2 AB 3 1 A 2 CD 2 1 C 6 AB 3 AB 4 CD 2 ch46 Vectors by Chtan FYKulai 11 Simple combinations 4 AB 1 C 1 BC 3 5 AC = 4 B A a c a c b d b d ch46 Vectors by Chtan FYKulai 12 Vector Geometry Consider this parallelogram Q OR b PQ P R a b O OP a RQ Opposite sides are Parallel OQ OP PQ a +b OQ OR RQ b +a a +b b + a OQ is known as the resultant of a and b ch46 Vectors by Chtan FYKulai 13 Resultant of Two Vectors □ Is the same, no matter which route is followed □ Use this to find vectors in geometrical figures ch46 Vectors by Chtan FYKulai 14 e.g.1 S is the Midpoint of PQ. Work out the vector . Q S P OS OS OP ½PQ = a + ½b R a b O ch46 Vectors by Chtan FYKulai 15 Alternatively S is the Midpoint of PQ. . Q S P Work out the vector OS OS OR RQ QS R a b O = b + a - ½b = ½b + a = a + ½b ch46 Vectors by Chtan FYKulai 16 e.g.2 C AC= p, AB = q p A M q Find BC M is the Midpoint of BC B BC = BA + AC = -q + p =p-q ch46 Vectors by Chtan FYKulai 17 e.g.3 C AC= p, AB = q p A M q Find BM M is the Midpoint of BC B BM = ½BC = ½(p – q) ch46 Vectors by Chtan FYKulai 18 e.g.4 C AC= p, AB = q p A M is the Midpoint of BC M q Find AM B AM = AB + ½BC = q + ½(p – q) = q +½p - ½q = ½q +½p ch46 Vectors = ½(q + p) by Chtan FYKulai = ½(p + q) 19 Alternatively C AC= p, AB = q p A M is the Midpoint of BC M q Find AM B AM = AC + ½CB = p + ½(q – p) = p +½q - ½p = ½p +½q ch46 Vectors = ½(p + q) by Chtan FYKulai 20 Distribution’s law : The scalar multiplication of a vector : + = + , ch46 Vectors > 0 < 0 by Chtan FYKulai 21 Other important facts : ℎ = ℎ ℎ + = ℎ + ch46 Vectors by Chtan FYKulai 22 A vector with the starting point from the origin point is called position vector. 位置向量 ch46 Vectors by Chtan FYKulai 23 Every vector can be expressed in terms of position vector. ch46 Vectors by Chtan FYKulai 24 e.g.5 2 −2 Given that = ,= 5 3 10 and also + = . Find 1 the values of . ch46 Vectors by Chtan FYKulai 25 e.g.6 Given that = − 4, = 3 − 2, and are parallel. Find the value of m. ch46 Vectors by Chtan FYKulai 26 e.g.7 2 3 = , = , a point 5 −2 1,4 . Find the coordinates of , then express point in terms of . ch46 Vectors by Chtan FYKulai 27 e.g.8 5 If 3,5 , = , find the −7 coordinates of . ch46 Vectors by Chtan FYKulai 28 e.g.9 Given that = 2 + , = 7 + + 4, and are parallel. Find the value of . ch46 Vectors by Chtan FYKulai 29 Magnitude of a vector 1 , 1 , 2 , 2 . = = − + − ch46 Vectors by Chtan FYKulai 30 , 0 = + Unit vector : = ∙ ch46 Vectors by Chtan FYKulai 31 e.g.10 Find the magnitude of the vectors : − = (b) = − ch46 Vectors by Chtan FYKulai 32 e.g.11 Find the unit vectors in e.g. 10 : − = (b) = − ch46 Vectors by Chtan FYKulai 33 Ratio theorem A P 1 a b B p 1 ch46 Vectors by Chtan FYKulai 34 e.g.12 M is the midpoint of AB, find b in terms of a, m . ch46 Vectors by Chtan FYKulai 35 e.g.13 2 3 6b 4a P divides AB into 2:3. Find OP in terms of a, b . ch46 Vectors by Chtan FYKulai 36 Application of vector in plane geometry e.g.14 A M X C In the diagram, CB=4CN, NA=5NX, M is the midpoint of AB. B CN u , BM v N (a) Express the following vectors in terms of u and v ; (i) NB (ii) NA ch46 Vectors by Chtan FYKulai 37 2 (b) Show that CX 4u v 5 (c) Calculate the value of (i) CX (ii) Area ACX CM Area ACM ch46 Vectors by Chtan FYKulai 38 Soln: (a) (i) CB CN NB NB CB CN 4CN CN 3CN 3u (ii) NA NB BA 3u 2v (b) 1 CX CN NX CN NA 5 1 8 2 2 u 3u 2v u v 4u v 5 5 5 5 ch46 Vectors by Chtan FYKulai 39 (c) (i) CM CB BM 4u v 2 CX CM 5 CX 2 CM 5 (ii) 1 CX h Area ACX CX 2 2 1 Area ACM CM 5 CM h 2 ch46 Vectors by Chtan FYKulai 40 e.g.15 M B A M and N are midpoints of AB, AC. N Prove that C ch46 Vectors 1 MN BC and MN // BC 2 by Chtan FYKulai 41 e.g.16 A 2a 1 6a B 1 K In the diagram K divides AD into 1:l, and divides BC into 1:k . l k O 2b C 6b D Express position vector OK in 2 formats. Find the values of k and l. ch46 Vectors by Chtan FYKulai 42 More exercises on this topic : 高级数学高二下册 Pg 33 Ex10g ch46 Vectors by Chtan FYKulai 43 Scalar product of two vectors If a and b are two non-zero vectors, θ is the angle between the vectors. Then , a b a b cos ch46 Vectors by Chtan FYKulai 44 Scalar product of vectors satisfying : Commutative law : a b b a Associative law : k a b a k b k a b Distributive law : a b c a b a c ch46 Vectors by Chtan FYKulai 45 e.g.17 Find the scalar product of the following 2 vectors : a 6 , b 5 , between is 60 ch46 Vectors by Chtan FYKulai 46 e.g.18 (a) If a b a b , find the angle between them. (b) If a 1, b 2, a k b and a k b are perpendicular, find k. ch46 Vectors by Chtan FYKulai 47 Scalar product (special cases) 1. Two perpendicular vectors a 0, b 0, a b a b 0 N.B. Unit vector for y-axis i j j i 0 Unit vector for x-axis ch46 Vectors by Chtan FYKulai 48 2. Two parallel vectors a 0, b 0, a // b a b a b N.B. i i 1 j j i i 1 j j ch46 Vectors by Chtan FYKulai 49 e.g.19 Given a 3, b 8, a b 2 14 , Find a b . Ans:[17/2] ch46 Vectors by Chtan FYKulai 50 Scalar product (dot product) The dot product can also be defined as the sum of the products of the components of each vector as : x1 x2 a , b y1 y2 a b x1 x2 y1 y2 ch46 Vectors by Chtan FYKulai 51 e.g.20 Given that 3 7 a ; b 4 1 Find (a) a b (b) angle between a and b . Ans: (a) 25 (b) 45° ch46 Vectors by Chtan FYKulai 52 Applications of Scalar product 高级数学高二下册 Pg 42 to pg43 Eg30 to eg 33 ch46 Vectors by Chtan FYKulai 53 More exercises on this topic : 高级数学高二下册 Pg 44 Ex10i Misc 10 ch46 Vectors by Chtan FYKulai 54 Miscellaneous Examples ch46 Vectors by Chtan FYKulai 55 e.g.21 Given that D, E, F are three midpoints of BC, CA, AB of a triangle ABC. Prove that AD, BE and CF are concurrent at a point G and AG BG CG 2 . GD GE GF ch46 Vectors by Chtan FYKulai 56 A Soln: From ratio theorem 1 d b c 2 1 e a c 2 1 f a b 2 ch46 Vectors B by Chtan FYKulai F G D E C 57 We select a point G on AD such that = . From ratio theorem, 1 2 1 1 g a b c a b c 3 3 2 3 Similarly, We select a G1 point on BE such that = . ch46 Vectors by Chtan FYKulai 58 1 g 1 a b c 3 Similarly, We select a G2 point on CF such that = . 1 g 2 a b c 3 ch46 Vectors by Chtan FYKulai 59 Because g1, g2, g are the same, G, G1, G2 are the same point G! G is on AD, BE and CF, hence AD, BE and CF intersect at G. And also = established. ch46 Vectors = by Chtan FYKulai = is 60 Centroid of a ∆ ch46 Vectors by Chtan FYKulai 61 ch46 Vectors by Chtan FYKulai 62 The end ch46 Vectors by Chtan FYKulai 63