MEP Practice Book ES19

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1 19 Vectors M rctice ook S Vectors nd Sclrs 1. Which of the following re vectors nd which re sclrs? Speed ccelertion Mss Velocity (e) Weight (f) Time 2. Use the points in the grid elow to find the vectors (i) (e) (f) (g) (h) (i) (ii) Wht is the reltionship etween (iii) nd Wht is equl to? 3. lot the positions of the points,,,,, nd F reltive to if = = 2 = 2 0 = 3 2 (e) 3 = 1 (f) F = If = c 1 = 1 0 = 2 3 2, find + + c 2 (e) 3 2 (f) + + c (g) 2 + 3c (h) 3 (i) c 104

2 M rctice ook S19 5. If = c 2 = 2 1 =, solve ech of the equtions elow to find the unknown vector, x. + x = x c = x + = 2x + = c (e) 2 x = 3c (f) 2 + x = + c (g) 2 + c + x = o (h) 3 + 2x = (i) 4 x = 2c 19.2 pplictions of Vectors 1. The wter in river flows t speed of 1 m/s. nn swims t speed of 2 m/s t right ngles to the nk. The river is 10 m wide. Find the time tken to rech the other side of the river, the distnce she trvels downstrem. 2. n eroplne trvels NW t speed of 400 m/s in still ir. If there is wind of speed 50 m/s heding directly west, find the resultnt speed nd direction of trvel of the plne. If the wind continues to hed west t speed of 50 m/s, find the required speed nd direction if the resultnt speed is 400 m/s in NW direction. 3. oy rows t 4 m/s cross river of width 27 m. The river flows t speed of 2.4 m/s. Find the direction in which the oy must row if he is to lnd on the other side directly opposite his strting point. lso find the time it tkes for him to rech the other side. 4. smll lock is held in position y 2 cles tht exert forces of mgnitude nd. Grvity exert downwrd force of 20 N. 45 The lock remins t rest. Find nd y using trigonometry. 20 N 5. The digrm shows the forces cting on cr, t rest, on slope. R 10 S Find the mgnitude of R nd S N 105

3 M rctice ook S Vectors nd Geometry 1. The digrm shows grid mde up of sets of eqully spced prllel lines. The vectors = nd = re shown. Write ech of the following in terms of nd. I H I J K L G F M GM (e) J (f) MN (g) K (h) (i) FI N (j) L (k) I (l) H 2. The digrm shows the prllelogrm in which =, =. Write the following vectors in terms of nd. (i) (ii) (iii) (iv) X is the midpoint of nd Y is the midpoint of. Find, in terms of nd, (i) X (ii) Y (iii) XY Wht is the reltionship etween XY nd? 3. In the digrm = nd 3 = The points nd re the mid-points of nd respectively. lculte the vector. lculte the vectors nd Use your nswers to prt to give two fcts out the lines nd. (SG) 106

4 M rctice ook S19 4. F F is regulr hexgon =, = Write down, in terms of nd, the vectors (i) (ii) F Write down one geometricl fct out nd F which could e deduced from your nswers to prt. (LN) 5. R S c RS nd STU re prllelogrms. U T is, S is, ST is c. Find, in terms of, nd c expressions in their simplest forms for T US X, where X is the midpoint of T. 1 + T 2 (LN) 107

5 19.3 M rctice ook S19 6. X In the digrm X is the point on such tht X = 3 X. Given tht = 8 nd = 4, express in terms of nd/or (i) (ii) X X (MG) 7. S R The digrm shows the position of points,, nd R with vectors nd cting long R nd, respectively. R = = S = 1 3 = 2 R y expressing S nd RS in terms of the vectors nd find the rtio S : SR nd explin the reltionship etween the points, S nd R. (SG) 108

6 M rctice ook S19 8. NT T SL c 4 In the digrm, is prllelogrm. = 4, = c, = Find, in terms of nd c, 1 4, = 4 3 (i) (ii) (iii) (iv) (Give your nswer in its simplest form.) Use your nswers to prt to explin why is trpezium. 9. F is regulr hexgon. =, =, Find the vector in terms of nd. = 2 X (R) = X rove tht is prllel to X. F 2 igrm NT ccurtely drwn (dexcel) 10. is tringle. X is the midpoint of. Y is the midpoint of. Z is the point on X such tht Z : ZX = 2 : 1 = 3, = 3 Find, in terms of nd, the vectors (i) Y (ii) X (iii) Z 3 Z X 3 Y Not drwn ccurtely, Z nd Y re on stright line. Find the rtio Z : ZY. 109 ()

7 M rctice ook S Further Work with Vectors 1. Write ech vector shown elow in component form, using components prllel nd perpendiculr to the dshed line For ech column vector elow, find its mgnitude nd drw digrm to show its direction, indicting the size of the relevnt ngles N 3. The three forces shown in the digrm ct t point. Find the mgnitude nd direction of its resultnt. 25 N N 15 N 4. The forces shown in the digrm re in equilirium, so tht their resultnt is the zero vector. Find F nd the ngle θ. θ 20 8 N F 110

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors