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Form 3 Mathematics: Vectors II Questions and Answers
The position vectors of points X and Y are x=2i+j-3k and y=3i+2j-2k respectively. Find XY.
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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.
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.
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.
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.
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
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.
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.
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.
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.
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
11.
The position vectors of points X and Y are x=2i+j-3k and y=3i+2j-2k respectively. Find XY.
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.
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.
14.
Given that OA=3i-2j+k and OB=4i+j-3k. Find the distance between points A and B to 2 decimal places.
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.
16.
Given that qi+ #1/3#j+#2/3# k is a unit vector, find q.
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.
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.
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.
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.
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.
22.
Given that P=2i-3j+k, Q=3i-4j-3k and R=3P+2Q, find the magnitude of R to 2 significant figures.
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
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.
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.
26.
Vectors r and s are such that r=7i+2j-k and s=-i+j-k. Find|r+s |.
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.
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