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Vectors in Two Dimensions (Cambridge O Level Additional Maths)

Exam Questions

1 hour10 questions
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1a
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Find the unit vector in the direction of open parentheses table row 5 row cell negative 12 end cell end table close parentheses.

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1b
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Given that open parentheses table row 4 row 1 end table close parentheses plus k open parentheses table row cell negative 2 end cell row 3 end table close parentheses equals r open parentheses table row cell negative 10 end cell row 5 end table close parentheses, find the value of each of the constants k and r.

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1c
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Relative to an origin O, the points A comma space B and C have position vectors bold p comma space 3 bold q minus bold p space and space 9 bold q minus 5 bold p respectively.

(i)
Find stack A B with rightwards arrow on top in terms of bold p and bold q.

[1]

(ii)
Find stack A C with rightwards arrow on top in terms of bold p and bold q.

[1]

(iii)
Explain why A comma space B space and space C all lie in a straight line.

[1]

(iv)
Find the ratio A B space colon space B C.

[1]

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1
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The parallelogram O A B C is such that stack O A with rightwards arrow on top space equals space bold a and stack space O C with rightwards arrow on top space equals space bold c bold space. The point D lies on O C such that O D space colon space D C = 1 colon space 2. The point E lies on A C such that A E space colon space E C =space 2 space colon space 1 space.

Show that stack O B with rightwards arrow on top space equals space k stack D E with rightwards arrow on top , where k is an integer to be found.

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2a
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In the diagram stack O P with rightwards arrow on top space equals space 2 bold b, stack O S with rightwards arrow on top space equals space 3 bold a, stack S R with rightwards arrow on top space equals space bold b and stack P Q with rightwards arrow on top space equals space bold a. The lines O R and Q S intersect at X.

Find stack O Q with rightwards arrow on top in terms of bold a and bold b.

0606-w20-qp-22-q9a

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2b
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Find stack Q S with rightwards arrow on top in terms of bold a and bold b.

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2c
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Given that stack Q X with rightwards arrow on top space equals mu stack Q S with rightwards arrow on top , find stack O X with rightwards arrow on top in terms of bold a comma space bold b space and space mu.

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2d
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Given that stack O X with rightwards arrow on top space equals space lambda stack O R with rightwards arrow on top, find stack O X with rightwards arrow on top in terms of bold a comma space bold b space and space lambda.

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2e
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Find the value of lambda and of mu.

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2f
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Find the value of fraction numerator Q X over denominator X S end fraction.

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2g
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Find the value of fraction numerator O R over denominator O X end fraction.

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3a
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q9-0606-w20-qp-13-additional-maths

The diagram shows the triangle O A C. The point B is the midpoint of O C. The point Y lies on A C such that O Y intersects A B at the point X where A X space colon X B space equals space 3 colon 1. It is given that stack O A with rightwards arrow on top space equals space a and stack O B with rightwards arrow on top space equals space b.

Find stack space O X with rightwards arrow on top spacein terms of a and b, giving your answer in its simplest form.

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3b
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Find AC in terms of a and b.

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3c
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Given that stack O Y with rightwards arrow on top space equals space h stack O X with rightwards arrow on top , find stack A Y with rightwards arrow on top in terms of a, b and h.

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3d
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Given that stack A Y with rightwards arrow on top space equals space m stack A C with rightwards arrow on top, find the value of h and of m.

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4a
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The vectors bold a and bold b are such that bold a space equals space alpha bold i plus bold space bold j and bold b space equals space 12 bold i plus bold space beta bold j.

Find the value of each of the constants alpha and beta such that 4 bold a minus bold b space equals space left parenthesis alpha plus 3 right parenthesis bold i minus 2 bold j.

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4b
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Hence find the unit vector in the direction of bold b minus 4 bold a.

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1a
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A particle P is initially at the point with position vector open parentheses table row 30 row 10 end table close parentheses and moves with a constant speed of 10 space ms to the power of negative 1 end exponent in the same direction as open parentheses table row cell negative 4 end cell row 3 end table close parentheses.

Find the position vector of P after t space straight s.

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1b
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As P starts moving, a particle Q starts to move such that its position vector afterspace t space straight s space is given by  open parentheses table row cell negative 80 end cell row 90 end table close parentheses plus t open parentheses table row 5 row 12 end table close parentheses.

Write down the speed of Q.

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1c
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Find the exact distance between P and Q when t space equals space 10, giving your answer in its simplest surd form.

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2a
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Relative to an origin O, the position vectors of the points A comma space B comma space C and D are

stack O A with rightwards arrow on top space equals open parentheses table row 6 row cell negative 5 end cell end table close parentheses comma space stack O B space with rightwards arrow on top equals open parentheses table row 10 row 3 end table close parentheses comma space stack O C with rightwards arrow on top space equals space open parentheses table row x row y end table close parentheses space a n d space stack O D with rightwards arrow on top space equals space open parentheses table row 12 row 7 end table close parentheses

Find the unit vector in the direction of stack A B with rightwards arrow on top.

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2b
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The point A is the mid‑point of B C. Find the value of x and of y.

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2c
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The point E lies on O D such that O E space colon space O D is 1 space colon space 1 plus lambda. Find the value of lambda such that stack B E with rightwards arrow on top is parallel to the x‑axis.

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3a
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In this question all distances are in km.
A ship P sails from a point A, which has position vector open parentheses table row 0 row 0 end table close parentheses, with a speed of 52 space kmh to the power of negative 1 end exponent in the direction of open parentheses table row cell negative 5 end cell row 12 end table close parentheses

Find the velocity vector of the ship.

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3b
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Write down the position vector of P at a time t hours after leaving A.

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3c
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At the same time that ship P sails from A, a ship Q sails from a point B, which has position vector open parentheses table row 12 row 8 end table close parentheses, with velocity vector open parentheses table row cell negative 25 end cell row 45 end table close parentheses space kmh to the power of negative 1 end exponent.
Write down the position vector of Q at a time t hours after leaving B.

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3d
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Using your answers to parts (b) and (c), find the displacement vector stack P Q with rightwards arrow on top at timespace t hours.

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3e
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Hence show that P Q space equals space square root of 34 t squared space minus 168 t plus 208 end root. 

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3f
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Find the value of t when P and Q are first 2 space km apart.

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4
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The position vectors of three points, A comma space B space and space C, relative to an origin O, are open parentheses table row cell negative 5 end cell row cell negative 7 end cell end table close parentheses comma space open parentheses table row 10 row cell negative 4 end cell end table close parentheses space and space open parentheses table row x row y end table close parentheses respectively. Given that stack A C with rightwards arrow on top = 4 stack B C with rightwards arrow on top, find the unit vector in the direction of stack O C with rightwards arrow on top.

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5a
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q8-0606-s20-qp-12-additional-maths

The diagram shows a triangle O A B such that stack O A with rightwards arrow on top space equals space bold a and stack O B with rightwards arrow on top space equals space bold b. The point P lies on O A such that O P equals 3 over 4 O A. The point Q is the mid-point of A B. The lines O B and P Q are extended to meet at the point R. Find, in terms of bold a and bold b,

Find stack A B with rightwards arrow on top

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5b
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Find stack P Q with rightwards arrow on top. Give your answer in its simplest form.

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5c
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It is given that n stack P Q with rightwards arrow on top space equals space stack Q R with rightwards arrow on top space and space stack B R with rightwards arrow on top space equals space k bold b, where n space and space k are positive constants.

Find stack Q R with rightwards arrow on top in terms of n comma space bold a space and space bold b.

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5d
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Find stack Q R with rightwards arrow on top in terms of k comma space bold a space and space bold b.

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5e
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Hence find the value of n and of k.

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