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# Form 3 Mathematics: Vectors II Questions and Answers

Form 3 Mathematics: Vectors II Questions and Answers

Lessons (**27**) * SHARE*

- 1.
In a parallelogram ABCD, AB=2a and AD=b. M is the midpoint of AB. AC cuts MD at X.
(i) Express AC in terms of a and b.
(ii) Given that AX=mAC and MX=nMD, where m and n are constants, find m and n.

9m 43s - 2.
In the figure below, E is the midpoint of BC, AD:DC=3:2 and F is the point of intersection of BD and AE.
(i) Given that AB=b and AC=c, express AE and BD in terms of b and c.
(ii) Given further that BF=tBD and AF=sAE, find the values of s and t.

14m 19s - 3.
Three points A, B and P are in straight line such that AP=tAB. Given that the coordinates of A, B and P are (3, 4), (8, 7) and (x, y) respectively, express x and y in terms of t.

4m 10s - 4.
The vectors p, q and y are expressed in terms of the vectors t and s as follows:
p= 3t + 2s
q= 5t – s
y=ht + (h-k)s
Where h and k are constants. Given that y=2p-3q, find the values of h and k.

4m 43s - 5.
Find the position vector of point R which divides line MN internally in the ratio 2:3.
Take the position vectors of M and N to be

6m 18s - 6.
In the diagram below, OABC is a parallelogram, OA=a and AB=b. N is a point on OA such that ON:NA = 1:2.
(a) Find:
(i) AC in terms of a and b.
(ii) BN in terms of a and b.
(b) The lines AC and BN intersect at X, AX=hAC and BX=kBN.
(i) By expressing OX in two ways, find the values of h and k.
(ii) Express OX in terms of a and b.

15m 48s - 7.
The points P, Q and R lie on a straight line. The position vectors of P and R are 2i + 3j + 13k and 5i – 3j + 4k respectively. Q divides PR internally in the ratio 2:1. Find the
(a) Position vector of Q.
(b) Distance of Q from the origin.

6m 13s - 8.
The figure below shows triangle OAB in which M divides OA in the ratio 2:3 and N divides OB in the ratio 4:1. AN and BM intersect at X.
(a) Given that OA=a and OB=b, express in terms of a and b:
(i) AN
(ii) BM
(b) If AX=sAN and BX=tBM, where s and t are constants, write two expressions for OX in terms of a, b, s and t.
Find the values of s. hence write OX in terms of a and b.

14m 34s - 9.
The position vectors for points P and Q are 4i + 3j + 2k and 3i – 6j + 6k respectively. Express vector PQ in terms of unit vectors i, j and k. Hence find the length of PQ, leaving your answer in simplified surd form.

3m 20s - 10.
The figure below shows a parallelogram OPQR with O as the origin, OP=P and OR=r. Point T divides RQ in the ratio 1:4. PT meets OQ as S.
(a) Express in terms of p and r, the vectors
(i) OQ
(ii) OT
(b) Vector OS can be expressed in two ways: mOQ or OT + nTP, where m and n are constants. Express OS in terms of
(i) m, p and r.
(ii) n, p and r.
Hence find the:
(iii) Value of m.
(iv) Ratio OS:SQ

14m 39s - 11.
The position vectors of points X and Y are x=2i+j-3k and y=3i+2j-2k respectively. Find XY.

1m 55s - 12.
Given that x=2i+j-2k, y=-3i+4j-k and z=-5i+3j+2k and that p=3x-y+2z, find the magnitude of vector p to 3 significant figures.

5m 25s - 13.
In the figure below, OP=p and OR=r. Vector OS=2r and OQ= #3/2#p.
(a) Express in terms of p and r:
(i) QR
(ii) PS
(b) The lines QR and PS intersect at K such that QK=mQR and PK=nPS, where m and n are scalars. Find two distinct expressions for OK in terms of p, r, m and n. Hence find the values of m and n.
(c) State the ratio PK:KS.

12m 15s - 14.
Given that OA=3i-2j+k and OB=4i+j-3k. Find the distance between points A and B to 2 decimal places.

2m 38s - 15.
Point T is the midpoint of a straight line AB. Given that the position vectors of A and T are i-j+k and 2i+1 #1/2# k respectively, find the position vector of B in terms of i, j and k.

5m 35s - 16.
Given that qi+ #1/3#j+#2/3# k is a unit vector, find q.

1m 57s - 17.
In the figure below, OQ=q and OR=r. point X divides OQ in the ratio 1:2 and Y divides OR in the ratio 3:4. Lines XR and YQ intersect at E.
(a) Express in terms of q and r.
(i) XR
(ii) YQ
(b) If XE=mXR and YE=nYQ, express OE in terms of
(i) r, q and m.
(ii) r, q and n.
(c) Using the results in (b) above, find the values of m and n.

12m 28s - 18.
Vector q has a magnitude of 7 and is parallel to vector p. given that p=3i-j+1 #1/2# k, express vector q in terms of i, j and k.

3m 11s - 19.
In the figure below, AB=p, AD=q, DE= #1/2#AB and BC= #2/3#BD.
(a) Find in terms of p and q, the vectors:
(i) BD
(ii) BC
(iii) CD
(iv) AC
(b) Given that AC=kCE, where k is a scaler, find:
(i) The value of k.
(ii) The ratio in which C divides AE.

10m 33s - 20.
The position vectors of points A and B are #((3),(-1),(-4))# and #((8),(-6),(6))# respectively. A point P divides AB in the ratio 2:3. Find the position vector of point P.

4m 38s - 21.
In the figure below, ABCD is a trapezium. AB is parallel to DC, diagonals AC and DB intersect at X and DC=2AB.
AB=a, DA=d, AX=kAC and DX=hDB, where h and k are constants.
(a) Find in terms of a and d:
(i) BC
(ii) AX
(iii) DX
(b) Determine the values of h and k.

9m 42s - 22.
Given that P=2i-3j+k, Q=3i-4j-3k and R=3P+2Q, find the magnitude of R to 2 significant figures.

3m 20s - 23.
In triangle OPQ below, OP=p, OQ=q. Point M lies on OP such that OM:MP=2:3 and point N lies on OQ such that ON:NQ=5:1. Line PN intersects line MQ at X.
(a) Express in terms of p and q:
(i) PN
(ii) QM
(b) Given that PX=kPN and QX=rQM, where k and r are scalers:
(i) Write two different expressions for OX in terms of p, q, k and r.
(ii) Find the values of k and r.
(iii) Determine the ratio in which X

12m 5s - 24.
In the figure below, OABC is trapezium. AB is parallel to OC and OC=5AB. D is a point on OC such that OD:DC=3:2.
(a) Given that OA=p and AB=q, express in terms of p and q:
(i) OB
(ii) AD
(iii) CB
(b) Lines OB and AD intersect at point X such that AX=kAD and OX=rOB, where k and r are scalars. Determine the values of k and r.

8m 58s - 25.
The position vectors of points F, G and H are f, g and h respectively. Point H divides FG in the ration 4:-1. Express h in terms of f and g.

3m 23s - 26.
Vectors r and s are such that r=7i+2j-k and s=-i+j-k.
Find|r+s |.

2m 4s - 27.
In the figure below, OA=a and OB=b. M is the mid-point of OA and AN:NB=2:1.
(a) Express in terms of a and b:
(i) BA
(ii) BN
(iii) ON
(b) Given that BX=hBM and OX=kON, determine the values of h and k.

9m 17s