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15 vector

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Nur Mina Alias
Nur Mina Alias

The document contains 12 problems related to vectors. The problems involve representing vectors in component form, adding and subtracting vectors, finding vectors given other vectors and scalar multiplication, determining if vectors are parallel or perpendicular, and using vectors to solve geometry problems. The solutions provide the requested vectors, scalars, or geometric relationships in terms of given information like vector components or other defined vectors.

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VECTOR PAPER 1 1. Given that 10 2 and . 7 4 p q     = = ÷  ÷     Find the value of m such that 4 5 p mq   − =  ÷ −  . [3 marks] Answer : ………………………….. 2. C A O B Answer : ………………………….. 3. Given that OABC is a parallelogram, 2 8 and OC 4OA i j= + = uuur uuur . Find the unit vector in the direction of OB uuur . [4 marks] Answer : …………………………... 1 DIAGRAM 1 Diagram 1 shows andOA x OB y= = uuur uuur . Express vector OC uuur in terms of andx y . [2 marks]
4. [4 marks] Answer : (a)………………………. (b)………………………. 5. Given that 12 .p ki j= − Find the values of k if 13=p . [2 marks] Answer : …………………………... 6. Given that a 8 and b 3 4 , find the vector a 2b.i j i j= + = − − [2 marks] Answer : …………………………... 2 O x Q (4, 3) P( ,2) y DIAGRAM 2 Diagram 2 shows two vectors, andOP OQ uuur uuur . Express (a) in the form x OP y    ÷   uuur , (b) in the formPQ uuur i jx y+ .
7. Given that 2 4 and . Find 3 1 OP OQ     = = ÷  ÷ −    uuur uuur (a) PQ uuur (b) QP uuur [3 marks] Answer : (a)………………………. (b)………………………. 8. DIAGRAM 3 In Diagram 3, PQRS is a parallelogram with PTR as a straight line. Given that 2 , 5 andPQ u PS v= = uuur uuur PR = 5 PT. Express, in terms of u and v , (a) PR uuur (b) QT uuur [4 marks] Answer : (a)………………………. (b)………………………. 9. If andm n are non-parallel and non-zero vectors, find the values of h and k if (3h + 5) m = (2h − 3k) n . [3 marks] Answer : …………………………... 3 Q P T S R
10. Given that andu v are parallel vectors. Find the values of k. [3 marks] Answer : …………………………... 11. Diagram 4 shows two vectors, OA uuur and AB uuur . Express a) OA uuur in the form x y    ÷   , b) AB uuur in the form i jx y+ . [2 marks] Answer : a) …………………………... b) …………………………... 4 2 3 4 u a kb v a b = − = + A(4, 3) x y 0 -5 B DIAGRAM 4
12. The points P, Q and R are collinear. Its is given that 4 2 and QR 3 (1 )PQ a b a k b= − = + + uuur uuur , where k is a constant. a) the value of k, b) the ratio of PQ : QR. [4 marks] Answer : a) k = ………………………... b) …………………………... PAPER 2 13. Given that 3 2 , 5 4 OP OQ −    = = ÷  ÷     uuur uuur and point ( 7, )R p− . Find (a) vector PQ uuur , [1 marks] (b) value of p if (i) P,Q and R are collinear [3 marks] (ii) 1 2 PQ PR= uuur uuur [3 marks] 5
14. In the Diagram 5, ABCD is a quadrilateral. AED and EFC is a straight line. B A C D E F DIAGRAM 5 Given that 1 20 , 8 , 25 24 , 4 AB x AE y DC x y AE AD= = = − = uuur uuur uuur and 3 5 EF EC= . (a) Express in terms of x and y (i) BD uuur (ii) EC uuur [3 marks] (b) Shows that the points B, F and D is a collinear. [3 marks] (c) If 2x = and 3y = , find BD [2 marks] 15. (a) Given that 8 and 3 4a pi j b i j= + = − + . Find the value of p if (i) anda b are parallel, (ii) 208a b+ = [5 marks] (b) APBQ is a parallelogram. 6 8 and 4 3 .AP i j AQ i j= + = + uuur uuur R is a midpoint for PQ . Find the following vectors in terms of andi j . (i) AR uuur (ii) QB QR+ uuur uuur [5 marks] Answer 6
1. m = 3 2. 3 2 2 x y− 3. 1 (3 4 ) 5 i j+ 4. (a) 1 2 −   ÷   (b) 5i j+ 5. k = 5 or k = −5 6. (2 9 )i j+ 7. (a) 2 4    ÷ −  (b) 2 4 −   ÷   8. (a) 2 5u v+ (b) 8 5 u v− + 9. 5 10 , 3 9 h k= − = − 10. 8 3 k = − 11. (a) 4 3 OA   =  ÷   uuur (b) 4 8i j− − 12. (a) 5 2 k = − (b) : 4:3PQ QR = 13. (a) 5 1 −   ÷ −  (b) (i) 3p = (ii) 3 , 7p p= = 14. (a) (i) 20 32x y− + % % (ii) 25x % (c) 104 15. (a) (i) 6p = − (ii) 5, 11p p= − = (b) (i) 11 5 2 i j+ % % (ii) 21 7 2 i j+ % % 7

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15 vector

  • 1. VECTOR PAPER 1 1. Given that 10 2 and . 7 4 p q     = = ÷  ÷     Find the value of m such that 4 5 p mq   − =  ÷ −  . [3 marks] Answer : ………………………….. 2. C A O B Answer : ………………………….. 3. Given that OABC is a parallelogram, 2 8 and OC 4OA i j= + = uuur uuur . Find the unit vector in the direction of OB uuur . [4 marks] Answer : …………………………... 1 DIAGRAM 1 Diagram 1 shows andOA x OB y= = uuur uuur . Express vector OC uuur in terms of andx y . [2 marks]
  • 2. 4. [4 marks] Answer : (a)………………………. (b)………………………. 5. Given that 12 .p ki j= − Find the values of k if 13=p . [2 marks] Answer : …………………………... 6. Given that a 8 and b 3 4 , find the vector a 2b.i j i j= + = − − [2 marks] Answer : …………………………... 2 O x Q (4, 3) P( ,2) y DIAGRAM 2 Diagram 2 shows two vectors, andOP OQ uuur uuur . Express (a) in the form x OP y    ÷   uuur , (b) in the formPQ uuur i jx y+ .
  • 3. 7. Given that 2 4 and . Find 3 1 OP OQ     = = ÷  ÷ −    uuur uuur (a) PQ uuur (b) QP uuur [3 marks] Answer : (a)………………………. (b)………………………. 8. DIAGRAM 3 In Diagram 3, PQRS is a parallelogram with PTR as a straight line. Given that 2 , 5 andPQ u PS v= = uuur uuur PR = 5 PT. Express, in terms of u and v , (a) PR uuur (b) QT uuur [4 marks] Answer : (a)………………………. (b)………………………. 9. If andm n are non-parallel and non-zero vectors, find the values of h and k if (3h + 5) m = (2h − 3k) n . [3 marks] Answer : …………………………... 3 Q P T S R
  • 4. 10. Given that andu v are parallel vectors. Find the values of k. [3 marks] Answer : …………………………... 11. Diagram 4 shows two vectors, OA uuur and AB uuur . Express a) OA uuur in the form x y    ÷   , b) AB uuur in the form i jx y+ . [2 marks] Answer : a) …………………………... b) …………………………... 4 2 3 4 u a kb v a b = − = + A(4, 3) x y 0 -5 B DIAGRAM 4
  • 5. 12. The points P, Q and R are collinear. Its is given that 4 2 and QR 3 (1 )PQ a b a k b= − = + + uuur uuur , where k is a constant. a) the value of k, b) the ratio of PQ : QR. [4 marks] Answer : a) k = ………………………... b) …………………………... PAPER 2 13. Given that 3 2 , 5 4 OP OQ −    = = ÷  ÷     uuur uuur and point ( 7, )R p− . Find (a) vector PQ uuur , [1 marks] (b) value of p if (i) P,Q and R are collinear [3 marks] (ii) 1 2 PQ PR= uuur uuur [3 marks] 5
  • 6. 14. In the Diagram 5, ABCD is a quadrilateral. AED and EFC is a straight line. B A C D E F DIAGRAM 5 Given that 1 20 , 8 , 25 24 , 4 AB x AE y DC x y AE AD= = = − = uuur uuur uuur and 3 5 EF EC= . (a) Express in terms of x and y (i) BD uuur (ii) EC uuur [3 marks] (b) Shows that the points B, F and D is a collinear. [3 marks] (c) If 2x = and 3y = , find BD [2 marks] 15. (a) Given that 8 and 3 4a pi j b i j= + = − + . Find the value of p if (i) anda b are parallel, (ii) 208a b+ = [5 marks] (b) APBQ is a parallelogram. 6 8 and 4 3 .AP i j AQ i j= + = + uuur uuur R is a midpoint for PQ . Find the following vectors in terms of andi j . (i) AR uuur (ii) QB QR+ uuur uuur [5 marks] Answer 6
  • 7. 1. m = 3 2. 3 2 2 x y− 3. 1 (3 4 ) 5 i j+ 4. (a) 1 2 −   ÷   (b) 5i j+ 5. k = 5 or k = −5 6. (2 9 )i j+ 7. (a) 2 4    ÷ −  (b) 2 4 −   ÷   8. (a) 2 5u v+ (b) 8 5 u v− + 9. 5 10 , 3 9 h k= − = − 10. 8 3 k = − 11. (a) 4 3 OA   =  ÷   uuur (b) 4 8i j− − 12. (a) 5 2 k = − (b) : 4:3PQ QR = 13. (a) 5 1 −   ÷ −  (b) (i) 3p = (ii) 3 , 7p p= = 14. (a) (i) 20 32x y− + % % (ii) 25x % (c) 104 15. (a) (i) 6p = − (ii) 5, 11p p= − = (b) (i) 11 5 2 i j+ % % (ii) 21 7 2 i j+ % % 7