St Andrew s Academy Mathematics Department Higher Mathematics VECTORS

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Jonas Snow
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1 St ndew s cdemy Mthemtics etment Highe Mthemtics VETORS St ndew's cdemy Mths et

2 Vectos sics 1. = nd = () Sketch the vectos nd. () Sketch the vectos nd. (c) Given u = +, sketch the vecto u. (d) Given v =, sketch the vecto v.. = nd = 5 7 () Sketch the vectos nd. ()Given u = +, sketch the vecto u. (c)given v =, sketch the vecto v.. = nd = Find () () (c) + (d) (e) ½ (f) (g) (h). Find nd in ech eution elow () () (c) Find, nd in ech of the following () () is the oint (,1,) nd is the oint (0,,). () Wite down the comonents of nd, the osition vectos of nd. () lculte (i) + (ii) (c) Find the vecto 7. is the oint (1,1,), Q is (,0,5) nd R is (1,1,0). () Wite down the comonents of, nd the osition vectos of, Q nd R. () Find the vectos (i) Q (ii) QR (iii) R St ndew's cdemy Mths et 0117

3 8. M is the oint (,,5), N is (1,1,0) nd R is (,,). () Wite down the comonents of m, n nd the osition vectos of M, N nd R. () Find the vectos (i) MN (ii) RN (iii) MR 9. QRS is llelogm with vetices (,,0), Q(7,,) nd R(8,5,). Find the coodintes of S. (Hint: Q = SR ). 10. is the oint (,1,), is (,1,9) nd is (0,1,). Given, find the coodintes of. 11. is the oint (,,), Q is (1,8,1) nd R is (5,,10). Given Q RS, find the coodintes of S. 1. () lculte the mgnitude of the vecto u =. () Find unit vecto llel to the vecto u. 1. () lculte the mgnitude of the vecto w = () Find unit vecto llel to w. 1. () lculte whee = 1 0 () Find unit vecto llel to. 15. The digm shows egul hexgon. (i) Wite down nothe vecto eul to () () (ii) Find vecto eul to () () (c) FE E F (d) F E (e) E EF (f) E St ndew's cdemy Mths et 0117

4 1. Use the digm oosite to nme the vecto tht eesents () () (c) t (d) s t (e) (f) t + (g) s + t s Q 17. The digm shows egul octgon. () Wite down nothe vecto eul to W R (i) W (ii) WV VU (iii) W SR RQ (iv) SR ST V S U T 18. The digm shows cuoid EFGH. E H () Exess in tems of u, v nd w. F (i) (ii) E (iii) FH (iv) H () Exess in tems of EFGH (i) u + v (ii) u w (iii) w u + v u G w v 19. In the tezium = nd is llel to. In tems of u nd v, wite down the vectos () () (c) (d) N v u _. N _ St ndew's cdemy Mths et 0117

5 Vectos 1. = 1 nd = 0. lculte () () (c) ( ). () Find the mgnitude of the vecto 1. () Find vecto llel to the vecto 0 which hs unit length.. is (0,,5), is (7,,9) nd is (1,1,17). Show tht, nd e colline stting the tio :.. QRS is llelogm with (,,0), Q(7,,) nd R(8,5,). Find the coodintes of S. 5. () is the oint (1,8,0) nd Q is (,,5). divides Q in the tio :. Find the coodintes of. () is (0,1,5) nd is (8,5,). Show tht, nd e colline.. n eolne flies in stight line t constnt seed. It tkes hous to fly fom to nd hous to fly fom to. Reltive to coodinte xes, is (0,1,) nd is (7,,1). Find the coodintes of. 7. u = i j + k nd v = i + j + 7 k. If u v find the vlue of. 8. Show tht the vectos = i j + k nd = i 7j k e eendicul. 9. tingle hs vetices (,1,9), (,,11) nd (7,8,1). Show tht this tingle is ightngled t. St ndew's cdemy Mths et

6 10. Thee oints, nd hve coodintes s shown. (1,0,) (1,8,) (5,,1) () Find the coodintes of if is llel nd eul in length to. () The oint E divides in the tio :1, find the coodintes of E. (c) ove tht E is eendicul to. 11. Use the digms to find the vlue of.. () 0 0 () 7 1. Wite down the vlue of tingle is fomed fom R(0,,1), S(1,5,) nd T(,1,). () Find the vectos RS nd RT. () Evlute RS. RT (c) Wht cn you deduce out he lines RS nd RT. 1.,, nd e the oints (1,,1), (1,,7), (0,,5) nd (1,,10) esectively. () Find the comonents of nd. () The vecto is eendicul to oth 1 nd. Find nd. St ndew's cdemy Mths et 0117

7 1 15. u = nd v 5. k 1 () Wite down the vectos u + v nd u v. () Given tht u + v nd u v e eendicul find k. 1. In the sue sed ymid oosite ll eight edges e of length 5 units. V Evlute.( + ). 17. Shown oosite is ightngled isosceles tingle. The two eul sides of the tingle hve length units. Find the vlue of k.(h + k + l). k h l T 18. In the digm oosite TOQR is ymid whose se OQR is homus of length 1 unit. OT nd ORT e euiltel tingles. () Evlute t.. () Given X is the midoint of Q, evlute t.x. t R x X Q O 19. The digm shows two vectos nd with nd. () Evlute (). (). (c). () Given = + evlute St ndew's cdemy Mths et

8 0. In the tezium = nd is llel to. In tems of u nd v, wite down the vectos () () (c) (d) N u _. N _ 1. EFGH is lleliied. In tems of u, v nd w find exessions fo () () H (c) (d) F (e) F u w v G H F E. () Fo the digm oosite find S nd T. () Hence clculte ngle TS. (1,,1) S(5,5,) T(,0,) E(1,0,). lculte the size of ngle FEG in the digm shown. F(1,1,0) G(0,,). nd e eesenttives of the vectos nd. nd nd ngle = θ. 1 7 () ove tht cos θ = 9 () Hence find the exct vlue of cos θ. St ndew's cdemy Mths et

9 5. In the digm = 15, = nd F = 8 () Wite down the coodintes of nd F () lculte the size of ngle F. z H E y G F x. The digm shows thee cuoids lced on to of ech othe. Two of the cuoids e eul in size 10 cm y cm y 5 cm. The thid cuoid is centlly lced on the othe two nd hs dimensions cm y cm y 5 cm. z y () Wite down the coodintes of, nd. () lculte the size of ngle x St ndew's cdemy Mths et

10 Vectos Revision 1. is the oint (,1,) nd is (10,,11). divides in the tio 1:. () Find the coodintes of. () Find. () hs coodintes (,1,) nd R hs coodintes (7,,1). Q divides R in the tio :. Find the coodintes of Q. () T is (1,0,) nd U is (10,,9). Show tht Q,T nd U e colline, stting the tio of QT:TU.. () is the oint (1,,) nd Q is the oint (,,). divides Q in the tio :. Find the coodintes of. () is the oint (u,1,1) nd is (11,7,). Given, nd e colline, Find u.. u = i + j nd v = 9i + j k. Show tht the vectos u nd v e eendicul. 5. u = 1 nd v =. 1 () Find the vectos u + v nd u v () Show tht the vectos u + v nd u v e eendicul.. hs coodintes (1,,), is (7,,1) nd is (9,7,1). Show tht tingle is ightngled t. 7. hs coodintes (1,1,1), Q is (,0,1) nd R is (7,,). lculte the size of ngle QR. Q R 8. is the oint (1,,), is (0,,) nd is (,0,). () Find the vectos nd () Hence clculte the size of ngle. z (,,9) y 9. The digm shows sue sed ymid of height 9 units. Sue O hs side of length 1 units. The coodintes of nd e (1,0,0) nd (,,9). lies on the yxis. () Wite down the coodintes of. () lculte the size of ngle. O (1,0,0) St ndew's cdemy Mths et x

11 10. Two vectos u nd v e such tht nd vu. v Given tht u.(u + v) =, show tht ngle θ = u U T 11. QRSTU is egul hexgon of side units. Q, QR nd RS eesent vectos, nd c esectively. S () Find the vlue of.( + c). () Wht cn you sy out the vectos nd ( + c). c Q R 1. The digm shows two vectos nd, with nd. () Evlute (i). (ii). () Evlute ( + ).( + ) 0 15 V 1. V is sue sed ymid. Exess in tems of, nd. () () (c) V Q 1. QRS TUVW is lleliied with vectos, nd c s shown. The length of ech side of the lleliied is units. ngle TUV = ngle TUR = () Exess U in tems of, nd c. () Show tht ngle UT = T S W c U R V St ndew's cdemy Mths et

π,π is the angle FROM a! TO b

π,π is the angle FROM a! TO b Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two