DDITIONL MTHEMTICS FORM 5 MODULE 7 VECTORS
VECTORS Contents Page 7.0 Conceptual Map 2 7.1 Introduction to Vector Practice 7.1 3 7.2 Multiplication of Vector y Scalar Practice 7.2.1 Practice 7.2.2 4 5 7.3 ddition nd Subtraction Practice 7.3 6 7-9 7.4 SPM Questions 10-12 ssessment test 13-14 nswers 15-16 1
7.0 CONCEPTUL MP VECTORS vector is a quantity that has a magnitude and a direction Magnitude Notation of Vectors or - a a a Multiplication of vector y a scalar ka = k a = a ; PQ = 2 = 2 ate a Two vectors are parallel if one of the vectors is the scalar multiple of the other vector 1 = 2 P PQ hence // Q PQ Parallel vectors Example (a) 2 + 3 = 5 (b) 4 a + 2 a = 6 a ddition and subtraction of vectors Non-parallel vectors (a) Triangle Law of vector dditions C = + C (b) Parallelogram Law of vector dditions S R C P PR = = PQ + QR PQ + PS Q Expression of a vector as the linear combination of a few vectors DC = DE + E + + C (c) Polygon Law F E D 2 F = C + C + CD + DE + EF
7.1 INTRODUCTIONS TO VECTOR Sharpen Your Skills 1 Find the magnitude and direction of vector. Distance of = 2 2 3 3 Magnitude Direction = 18 = 4.24 unit = 4.24 unit = North-East PRCTICE 7.1 Write the notation of vector and find the magnitude and direction a M D e E b K d C Q F f P G H c Notation Magnitude Direction 5.099 unit North-East 3
7.2 MULTIPLICTION OF VECTOR Y SCLR Sharpen Your Skills 2 a P 1. Express and PQ as a scalar product of Solution = 2 PQ = a 1 1 2 a a PRCTICE 7.2.1 1. In the figure, CEG is a parallelogram. It is given that H = a, J = c. Points, D, F, H are the midpoints of C, CE, EG and G Q respectively. Find each of the following vectors in terms of a, = b and c b and a H c G b C J F D E (a) E = (b) C = (c) G = (d) (e) = F = HD Sharpen Your Skills 3 Given = = u and C u, C = = 4u 4u, shows that, and C are collinear C = 4 C Point is common point., and C are collinear 4
PRCTICE 7.2.2 1. S 4 w R Find the parallel vector of each of the following 2u 3u P 3v Q D 6u C 3v 2. Figure below shows a trapezium CD in which that DC = 6u, = 4 cm and DC and DC = 3 cm, express are parallel. Given in terms of u. 3. Given that collinear. = 4 x and C = 6 x. Shows that the point, and C are 5
4. Given that (2h 3) a = (k + 5) b, find the value of h and k if b are non- parallel and non-zero vectors. a and 7.3 DDITIONS ND SUTRTIONS VECTORS Sharpen Your Skills 4 1. a + 2 a +2 a Find the sum of the vectors Sum of vectors = 10 a 2. P x Q 2y R P Q R P R P Q Q R 3. a b C a = x y b C C C = a ( b) = a b 6
PRCTICE 7.3 1. C D In the above figure, point D is on C, (a) Write (b) (i) (ii) D D in term of C in term of D and DC Find the resultant vector for C and + C 2. E D C In the figure, CD is a parallelogram which diagonals C and D intersect each other at point E. Find the resultant vector for each of the following. (a) D (c) C C (b) CE ED (d) E DE 7
3. E D C x y ased on the above figure, CDE is a straight line with C CD DE. Given that 3 x 2 y and C x y. Express each of the following vectors in terms of a and / or b 3x 2y (a) ED (a) C (a) D 4. F E C D In the above figure, CDEF is a regular hexogen. Find the resultant vector for each of the following (a) C CD (b) C DE CD (c) CD EF C DE F 8
5. a b C In the above figure, a and C b. If Q is a point above C so that Q : QC 1:3, express Q in terms of a and b. 9
7.4 SPM QUESTION 1. SPM 2003 P = 2a + 3b q = 4a b r= ha + (h k)b, where h and k are constants Use the above information to find the values of h and k when r = 3p 2q [3 marks] 2. Diagram shows a parallelogram CD with ED as a straight line. D C E Given that 6 p, D 4 p and DE = 2E, express in terms of p and q (a) D (b) EC [ 4 marks] 10
3. SPM 2004 Diagram below shows triangles O. The straight line P intersects the 1 straight line OQ at R. It is given that OP O, 3 1 Q, OP 6x and OP 2 y 4 Q R O P (a) Express in terms of (i) P, x and/or y : (ii) OQ [4 marks] (b) (i) Given that R h P, state R in terms of h, x and y (c) (ii) Given that RQ k OQ, state RQ in terms of k, x and y [2 marks] Using R and RQ from (b), find the value of h and of k [4 marks] 11
4. SPM 2005 In diagram, CD is a quadrilateral. ED and EFC are straight lines. D E F C It is given that 20 x, 1 E 8 Y, DC 25x 24 y, E D and EF 4 (a) Express in terms of (i) D, x and/or y : (ii) EC [3 marks] (b) Show that the points, F and D are collinear [2 marks] 3 5 EC (c) If x 2and y 3, find D [2 marks] 12
SSESSMENT C G H 1. In the above figure, GH : = 3 : 10 and GH is parallel to. If = 10 a, find GH in terms of a. [ 2 Marks] 2. T S U R P Q In the above figure PQRSTU is a regular hexagon. Express PQ + PT - RS single vector. as a 13
3. Given = (k + 1) a and C = 2 b = 3 a. Find the value of k. b. If, and C are collinear, = C and [ 3 Marks] 4. Given that OC is a rectangle where O = 6 cm and OC = 5cm. If O = a ando = b, find (a) C in terms of a and b [ 3 Marks] (b) a b 5. In OPQ, OP = p and OQ = that M is the midpoint of OT, express PM q. T is a point on PQ where PT : TQ=2 : 1. Given in terms of p and q. [ 3 Marks] 14
NSWERS PRCTICE 7.1 notation magnitude direction CD 2.828 North west HG 3 North FE 3.162 South-west MK 2 East QP 2.236 South-west PRCTICE 7.2.1 PRCTICE 7.2.2 1. (a) 2 c 1. PS (b) 2 b 2. 8u QR (c ) 2 a 3. h = 3 2 (d) -2 a (e) 2 b, k= -5 PRCTICE 7.3 1. (a) (i) DC (b) (ii) + C 4. (a) D + DC (b) E (c ) 0 2. (a) D 5. (b) (c) (d) CD D 1 4 b + a 3. (a) y x (b) 4 x + y (c) 5 x 15
PST YERS QUESTION 1. SPM 2004 = (a) (i) P 2 y + 6 x (ii) OQ 3 = 2 y + 9 2 x (b) (i) R = h( 6 x - 2 y ) (ii) RQ 9 = k( 2 x + 3 2 y ) (c) k = 1 3, h = 1 2 2. h = -2, h = -13 3. (a) 6 p 4q (b) 2. SPM 2005 8 2 p q 3 (a) (i) D = (c) D = 104 (ii) EC = 25 x 20 x + 32 y SSESSMENT 1. 3 a 2. PR 3. k = 5 4. (a) 5. b - 2 a (b) 13 units 1 3 q + 5 6 p 16
DDITIONL MTHEMTICS FORM 5 MODULE 8 VECTORS 17
VECTORS Contents Page 8.0 Conceptual Map 2 8.1 Vector In The Cartesian Coordinates 3-7 8.2 SPM Questions 8 10 8.3 ssessment test 11-12 nswers 13-14 18
8.0 CONCEPTUL MP VECTOR VECTOR IN THE CRTESIN COORDINTE y i j 1,1 r x i y j x i = 1 0, j 0 1 x vector r x i y j y The Magnitude r x y 2 2 The Unit Vector r r 1 ( x i y j) 2 2 r x y For any two points P and Q, the position vector of Q relative to P is defined by PQ OQ OP, where O is the origin. 19
8.1 VECTOR IN THE CRTESIN COORDINTES 1. State the following vector in terms in i and j and also in Cartesian coordinates Example j 5 4 3 2 1 0 Exercise 1 p 2 P 3 4 5 i Solutions 2 O 2 i 0 0 O 3 j 3 OP p 3i 4 j 3 4 Solutions j (a) OP = (b) OQ Q R 4 3 2 1-3 -2-1 O -2 1 P T 1 2 3 4-1 W S i (c) OR (d) OS (e) OT (f) OW 20
2. Find the magnitude for each of the vectors Example 3 i 2 j = 3 2 2 2 13 unit (a) 2 i 5 j (b) 5i12 j (c) i j = 3. Find the magnitude and unit vector for each of the following Example r 3i 4 j Solution : Magnitude, r 3 4 2 2 = 5 1 unit vector, r, (4 i 3 j) 5 (a) r 2 i 6 j (b) 6 a 3 1 (c) h 2 21
4. Given that (a) a b c a 3i 2 j, b 2 i 5 j and c 4 i 3 j. Simplify of the following (b) a 2b 3c 5. Given that a 2 i3 j, b i4 j and c 2 i 5 j. Determine the unit vector in the same direction with vector 2 a 3b c. 6. If O i3 j, O 2 j and C 4 i 3 j. Find (a) OC (b) C 22
Example Given that a 6 i7 j and b p i 2 j.find the possible value (or values) of p for each of the following cases. (a) a and b are parallel. (b) a b 7. Given r 4 i ( k 3) j and s ( k 4) i 14 j. If r is parallel to find (a) the value of k (b) s 3r for the positive values of k s, Solutions a k b 6 i 7 j k( p i 2 j) 6 i 7 j kp i 2k j y equating the coefficients of 6 = kp (1) i y equating the coefficients of 7 = 2k.(2) k = 7 3 1 2 2 Subtitute k = 7 into (1) 2 j 6 = 7 2 p p = 12 1 5 7 7 (b) a b 6 7 p 2 2 2 2 2 2 85 p 4 p² +4 = 85 p² = 81 p = 9 23
Example O is the origin, P and Q are two points such 2 k 3k 2 that OP OQ and -6 3. If the points O, P and Q are collinear, find the possible values of k. 8. Given that PQ 6 i 8 j and PR 2 i b j.if P, Q and R are collinear, find the value of b. Solutions Since the points O, P and Q are collinear, OP moq ( m is a constant) 2 2 k 3 k m 3-6 2 2 m( k 3 k) 3-6m m(k² - 3k) =2..(1) -6m = -3...(2) m = 1 2 From (1): 1 (k² - 3k) =2 2 k² - 3k = 4 k² - 3k- 4 = 0 (k 4 )( k + 1 ) = 0 k = 4 or -1 24
8.2 SPM QUESTIONS SPM 2003/no. 12 / paper 1. 1. Diagram 2 shows two vectors, OP and QO. Q(-8, 4) y P(5, 3) O Express x (a) OP in the form, y (b) OQ in the form xi + yj. x [ 2 marks] SPM 2005 / no. 15 / Paper 1 2. Diagram 5 shows a parallelogram, OPQR, drawn on a Cartesian plane. y Q R P O x It is given that OP = 6i + 4j and PQ -4i 5j. Find PR. [3 marks] 25
SPM 2004 3. Given that O(0, 0), ( -3, 4) and ( 2, 16), find in terms of the unit vectors, (a) (b) the unit vector in the direction of [ 4 marks] i and j SPM 2004 4. Given that ( -2, 6), ( 4, 2) and C( m, P), find the value of m and of p such that 2 C 10 i 12 j. [ 4 marks] 26
SPM 2003 5. 5 2 k Give that, O and CD 7 3 5, find (a) the coordinates of [ 2 marks] (b) the unit vector in the direction of O [ 2 marka] (c) the value of k, if CD is parallel to [ 2 marks] 27
8.3 SSESSMENT TEST 1. Find the magnitude and unit vector of a = 6 i 8 j [2 marks] 2. Find the unit vector in the direction of OP 12 5 [2 marks] 3. Given that u 3i m j and v k i 3 j, find the value of m and of k if u mv 11i 4 j [3 marks] 5 6 k 4. Three points, and C are such that O, O and OC 6 4 2, where O is the origin. Find the value of k if the points, and C are collinear [4 marks] 28
5. It is given that a 2 i 2 j, b i 2 j, P(1, -3) and Q(5, 2). If h and k are constants, find (a) the value of h and k PQ h a k b, where [4 marks] (b) the unit vector in the direction of PQ. [2 marks] 6. PQRS is a parallelogram. M is the midpoint of QR. Given that PQ 3i j and PM 4 i 2 j, find (a) (i) QR (ii) PR [4 marks] (b) the length of PS [2 marks] 7 11 7.Given that u and v, find the possible value(s) of k for each of the 9 k 3 following cases. (a) u and v are parallel [3 marks] (b) u v [3 marks] 29
NSWERS Exercises ssesment 3 4 1. (a) OP i 3 j 1. i j 5 5 12 13 (b) OQ 3i 2 j 2. 5 13 (c) ( d) (e) (f) OR i 2 j 3 m =2, k = 4 OS 2 i 2 j 4 k = 7 1 OT 4 i 5 (a) k =3, h = 2 OW 2 j 1 (b) (4 i 5 j ) 41 2. (a) 13 units 6 (a) (i) (b) 13 units 2 i 2 j (ii) 5i 3 j ( c) 2 units (b) 2.828 units 3. (a) 40 units, 1 1 (2 i 6 j ) 7 k = 17 40 7 (b) 45 units, 1 6 45 3 k = 0 or 6 (c) 5 units, 1 1 5 2 4. (a) 9 i 6 j (b) 19 i17 j 5. 1 ( 5 i 23 j) 554 6. (a) 4 i 5 j (b) 89 30
7. (a) k = -11, 4 (b) 6 units 8. b = 8 3 SPM QUESTIONS 1. (a) 2. (b) 3. (a) 10 i j (b) 5 OP 3 OQ 8i 4 j 5 2 1 5 13 12 4. m = 6, p = -2 5. (a) 3,4 1 3 (b) 5 4 (c) k = 25 7 31