Vectors Practice Answers

1 Worksheet 1 – Vectors 1. Answer (a) magnitude of 𝐴𝐵#####⃗ = &𝐴𝐵#####⃗& = √3! + 4! = 5 (b) magnitude of 𝐵𝐶#####⃗ = &𝐵𝐶#####⃗& = -(−5)! + (−12)! = 13 (c) magnitude of 𝐶𝐷#####⃗ = &𝐶𝐷#####⃗& = -3! + (−5)! = 5.83 (3.s.f) 2. Answer (a) 𝑂𝐴#####⃗ = 7−4 6 9 𝑂𝐵#####⃗ = : −1 𝑦 − 2< (b) 𝑂𝐶#####⃗ = 𝑂𝐴#####⃗ + 𝐴𝐶#####⃗ = 7−4 6 9 + 7 4 −109 = 7 0 −49 C = (0, −4) (c) 𝐴𝑂#####⃗ = −𝑂𝐴#####⃗ = 7 4 −69 (d) 𝐴𝐵#####⃗ = 𝐴𝑂#####⃗ + 𝑂𝐵#####⃗ = 7 4 −69 + : −1 𝑦 − 2< = : 3 𝑦 − 8< 𝐴𝐵#####⃗ = 𝑘𝐴𝐶#####⃗ : 3 𝑦 − 8< = 𝑘 7 4 −109 : 3 𝑦 − 8< = 7 4𝑘 −10𝑘9 3 = 4𝑘 𝑘 = " # 𝑦 − 8 = −10 7 " #9 𝑦 = −7.5 + 8 = $ ! 3. Answer (a) &𝑃𝑄#####⃗& = √3! + 4! = 5 (b) 𝑂𝑄######⃗ = 𝑂𝑃#####⃗ + 𝑃𝑄#####⃗

2 𝑂𝑃#####⃗ = 70 39 𝑂𝑄######⃗ = 70 39 + 7−3 4 9 = 7−3 7 9 𝑄 = (−3, 7) (c) 𝑂𝑅#####⃗ = 7𝑥 09 𝑃𝑅#####⃗ = 𝑃𝑂#####⃗ + 𝑂𝑅#####⃗ = − 70 39 + 7𝑥 09 = 7 𝑥 −39 (d) 𝑃𝑄#####⃗ = 𝑘𝑃𝑅#####⃗ 7−3 4 9 = 𝑘 7 𝑥 −39 7−3 4 9 = 7 𝑥𝑘 −3𝑘9 −3 = 𝑥𝑘 4 = −3𝑘 𝑘 = − # " (e) −3 = 𝑥𝑘 −3 = 𝑥 7− # "9 𝑥 = % # 𝑂𝑅#####⃗ = F2 $ # 0 G 4. Answer (a) 𝑂𝐴#####⃗ = 75 29 and 𝑂𝐵#####⃗ = 7 6 −39 𝐴𝐵#####⃗ = 𝐴𝑂#####⃗ + 𝑂𝐵#####⃗ = − 75 29 + 7 6 −39 = 7−5 −29 + 7 6 −39 = 7 1 −59 &𝐴𝐵#####⃗& = -1! + (−5)! = 5.10 units

3 (b) 𝑚 of 𝐴𝐵 = &"&! '&( = −5 𝑦 = 𝑚𝑥 + 𝑐 2 = −5(5) + 𝑐 𝑐 = 27 𝑦 = −5𝑥 + 27 (c) 𝐴𝐵#####⃗ = 2𝐴𝐶#####⃗ = 2𝐵𝐶#####⃗ $ ! 𝐴𝐵#####⃗ = 𝐴𝐶#####⃗ = 𝐵𝐶#####⃗ 𝐴𝐶#####⃗ = $ ! 7 1 −59 = L $ ! − ( ! M 𝑂𝐶#####⃗ = 𝑂𝐴#####⃗ + 𝐴𝐶#####⃗ = 75 29 + L $ ! − ( ! M = L 5 $ ! − $ ! M 𝐶 = (5.5, −0.5) 5. 𝑄𝑋#####⃗ = 𝑄𝑂######⃗ + 𝑂𝑋#####⃗ 𝑄𝑂######⃗ = −𝑂𝑄######⃗ = −𝑞 𝑂𝑋#####⃗ = " ( 𝑂𝑃#####⃗ = " ( 𝑝 𝑄𝑋#####⃗ = −𝑞 + + " ( 𝑝 6. Answer (a) Answer (i) 𝑂𝑅#####⃗ = 𝑂𝑃#####⃗ + 𝑃𝑅#####⃗ 𝑃𝑅#####⃗ = 2𝑃𝑄#####⃗ = 2𝑞 𝑂𝑅#####⃗ = 𝑝 + 2𝑞 (ii) 𝑇𝑅#####⃗ = 𝑇𝑂#####⃗ + 𝑂𝑅#####⃗ 𝑇𝑂#####⃗ = 3𝑃𝑂#####⃗ = −3𝑂𝑃#####⃗ = −3𝑝 𝑇𝑅#####⃗ = −3𝑞 + 𝑝 + 2𝑞 = −2𝑝 + 2𝑞 (iii) 𝑄𝑆#####⃗ = 𝑄𝑅#####⃗ + 𝑅𝑆#####⃗

4 𝑄𝑅#####⃗ = 𝑞 (since Q is midpoint of PR, PQ = QR) 𝑅𝑆#####⃗ = $ ' 𝑅𝑇#####⃗ = − $ ' 𝑇𝑅#####⃗ = − $ ' (−2𝑝 + 2𝑞) 𝑄𝑆#####⃗ = 𝑞 − $ ' (−2𝑝 + 2𝑞) = 𝑞 + ! ' 𝑝 − ! ' 𝑞 = ! ' 𝑝 + # ' 𝑞 = $ " (𝑝 + 2𝑞) (b) 𝑄𝑆#####⃗ = 𝑘𝑂𝑅#####⃗ 𝑄𝑆#####⃗ = $ " 𝑂𝑅#####⃗ Since they are a factor of one another, they are parallel. 7. Answer 8. Answer

5 9. Answer

6 Worksheet 2 – Vectors 1. Answer 2. Answer

7 3.

8 4. Answer (a) Answer (i) 𝐴𝐶#####⃗ = 𝐴𝐷#####⃗ + 𝐷𝐶#####⃗ = 4𝑞 + " ( 𝐴𝐵#####⃗ = 4𝑞 + " ( (5𝑝) = 4𝑞 + 3𝑝 (ii) 𝐵𝑋#####⃗ = 𝐵𝐴#####⃗ + 𝐴𝑋#####⃗ = −5𝑝 + ( ) 𝐴𝐶#####⃗ = −5𝑝 + ( ) (4𝑞 + 3𝑝 ) = −5𝑝 + !*+ ) + $(, ) = ( ! 𝑞 − !(, ) = ( ) (4𝑞 − 5𝑝) (iii) 𝑋𝐷#####⃗ = 𝑋𝐴#####⃗ + 𝐴𝐷#####⃗ = − ( ) 𝐴𝐶#####⃗ + 4𝑞

9 = − ( ) (4𝑞 + 3𝑝 ) + 4𝑞 = " ! 𝑞 − $( ) 𝑝 = " ) (4𝑞 − 5𝑝) (b) 𝐵𝑋#####⃗ = 𝑘𝑋𝐷#####⃗ ( ) (4𝑞 − 5𝑝) = " ) 𝑘(4𝑞 − 5𝑝) ( ) = " ) 𝑘 𝑘 = ( " 𝐵𝑋#####⃗ = ( " 𝑋𝐷#####⃗ Thus B, X and D lie on the same straight line. 5. Answer (a) Answer (i) 𝐴𝐵#####⃗ = 𝐴𝑂#####⃗ + 𝑂𝐵#####⃗ = − 7−1 5 9 + 7 2 −39 = 7 3 −89 (ii) √3! + 8! = 8.54 units (3.s.f.) (iii) 𝑂𝐶#####⃗ = 𝑂𝐴#####⃗ + 𝐴𝐶#####⃗ = 7−1