IB Mathemathics SL Review of Chapter 13: VectorsProblems Based on IB Mathematics SL Section A

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1. Given that u = C~)and v = (-i), find (u - v}u .

- -2. Given that IOAI = 3, lOBI = 5, and OA~OB = 7, (a) fmd the measure of AOB; (b) Then

;._ sketch MOB and find IABI·

p. Given that u = (-;) and v = (;), find .Ca) w such that u + v + W = 0; (b) the value of the

constant k such that (u + kv) is parallel to the x-axis.4. Two vectors oflength 5 and 6 units respectively make an angle of 50° with each other. Find

the length of the resultant.

'5. Given that a = (!) and b = (~), find (i) the vector d such that d = a + b ; (ii) a vector c

such that c bisects the acute angle between a and b.

6. Given that u = (121) and v = (-;), find the value of k such that (u + kv) is perpendicular to v.

-7. Given that u = (i) and v = C~),and that point A is at (-4,-2) with AB = u and

CA = v, (a) find the coordinates of points Band C; (b) find the area of triangle ABC.

8. Two vectors are given by p = (~) and q = (k ~8) ,k E R Find the value of k for whichp and

q are mutually perpendicular.

9. The vectors u and v are given by u = Ci) and v = (~).

(a) Find Iu - vi. (b) Find constants x andy such that x u + y v = (2°2).

- -10. The positions of points A andB are given by OA = (~) and OB = C;) . (a) Find the

distance between A and B . (b) Find the size of angle AOB to the nearest tenth of a degree.11. The diagram shows a parallelogram LMNO. The points Nand M y Mhave coordinates (15,8) and (23,14) respectively. The diagram is

not to scale. Find (a) ON and NM; (b) ILMI and [Ol.] , 0 x12. The vector v is given by v = 5i + 4j. The line L is perpendicular to v and contains the point

with position vector 3; + 2j. Find an equation for L in the form ax + by = c.

13. Vectors u and v are given by u = cj) and v = C~). (a) Find u + v; (b) Find a unit vector

perpendicular to (u + v) .14. Given two vectors a = 2; - j and b = 4; + 2j, find (a) la I x Ihl; (b) (b - a)oa .15. Given that u = i + 4j and v = 3; - 7j, find the angle between the vectors u and v , giving your

answer correct to the nearest tenth of a degree.

16. Triangle OPQ has one vertex at the origin O. Vertices P and Q are.given by the position-;> -;>

vectors OP = G) and OQ = C~) . Find the size of POQ, giving your answer correct to the

nearest tenth of a degree.

17. Given the ~ectors u = o and v = (~) , k E R, find a value for which the angle between uand v is 60°, giving your answer correct to two decimal places.

18. The diagram shows the vectors (124)and C;). Find ~o

;.,. the value of a, giving your answer correct to 1 a ..,. _j decimal place. -----~.---!19. ABeD is a rectangle and 0 is the midpoint of [AD].

-;> -;> :f<¢::(A BExpress each of the following in terms of OB and OC:-;> -;. -;.

(a) BC (b) OD (c) DC20. The vectors i ,j are unit vectors along the x-axis andy-axis respectively. The vectors

u = - 5i + 3 j and v =4 i + 7 j are given. (a) Find u + 3 v in terms oi i andj .A vector w has the same direction as u +·3 v , and has a magnitude of 100.(b) Find w in terms of i and j .

21. The vectors u , v are given by u = - 5i + 3 j and v = 4 i + 7 j .Find scalars a , b such that a(u + v) = 30i + (2 - b)j.

22. Find a vector equation of the line passing through (-2,-10) and (3,-6). Give your answer inthe form r= p + td , where tE R

23. Find the projection of v = 2i + 3j in the direction of w = 4i - 2j.-;> -;.

24. Find the projection of AB in the direction of CD for A(3,2), B(-7,3), C(9,0), D(-1,-5).25.26.

Find the magnitude of the projection of v = 6i + 2j in the direction of w = 3i + j.Copy the diagram onto graphing paper, and mark on ···~.t:···I······ ··t· · ~ i ..C,· : Ithe copy the following po~s: -;> " l : i(a) the point D such that AD = 3 AC ; ,j J

-;> -;> , A l Bl. (b) the point P such thatAP = AB - 2 AC ; !

-;>

-;>

-;> -;.

27.(c) the point Q such that AQ is the projection of the vector AC in the direction of AB.:~:(~~r::m:::,~{~rE:jof intersection of the lines with vector equations

28. Find the scalar projection of the vector v = 4i - 3j in the direction of the vector w = 7i + 24j .29. The vector equations for the two lines LI and L2 are r = 3i + 2j + s(2i - 6j)and

r = i + 3j + t (2 i - 2j). Find the cosine of the angle between the lines L I and L2.30. A particle moves along a line with constant velocity. The particle's initial position is

A(-7, 1,3). After two seconds, the particle is at B(5, -3, -1). (a) Find (i) velocity-;>

vector AB; (ii) the speed of the particle. (b) Write down the equation of (AB).

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Answers to illMathemathics SL Review of Chapter 13: VectorsProblems Based on illMathematics SL Section A

1. 58

2. (a) ACE ~ 62.2'-(b) IABI = 2v'S ~ 4.47

A "

~Bo S'1 SS-.(a) (7); (b) k=-"2

]4. 9.98 (3 s.f.)

5. (i) (!) (H)o. (IT any vector of form(ta)41

.6. - 74

7. (a) B(S,-l); QO,3); (b) ~1

8. 69. (a) 10 (b) x = 3; y = 110. (a) SV2 (b) 72.3°- -1L (a) ON = ei) ; NM = (~)

(b) ILMI = 17; lOLl = 1012. 5x + 4y = 23

14. (a) 10 (b) 1IS. 142.8°16, 100.6°

17. -0.43 or -8.3018. 105.8° - -(b) 1OC - lOB

2 219. (a) OC - OB- -I 1(c) "20C + "20B

20. (a) 7i + 24j (b) 28i + 96j21. a = -30; b = 302

22. r = C;o) + t(~)or r = e6) + t(~)23 2, I.

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24 - 38. 19 .. 51-5J

16 _8VlO2S. --or --VlO S

26. ~ "~ K' -.. -1- 'l~ ++..._.! 1 I\. ~. 1 i !

+-1 ';

++- -+ ~" ~ + .. l; ;f'"·1·+· -t- -j-- -+ ,~ ~ .

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