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Vectors (H) - Questions
Q1.
The vector a and the vector b are shown on the grid.
(a) On the grid, draw and label vector –2a(1)
(b) Work out a + 2b as a column vector.
(2)
(Total for question = 3 marks)
Q2.
M is the point such that BM : MC is 1 : 2
Here is Charlie's method to find in terms of a and b.
(a) Evaluate Charlie's method.
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(1)Martin expands (2x + 1)(2x − 3)(3x + 2)
He gets 12x3 − 4x2 − 17x + 6
(b) Explain why Martin's solution cannot be correct.
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(1)
(Total for question = 2 marks)
Q3.
OAYB is a quadrilateral.
(a) Express in terms of a and b.
..............................................................................................................................................(1)
X is the point on AB such that AX : XB = 1 : 2
and = 5a – b
*(b) Prove that
(4)(Total for Question is 5 marks)
Q4.
APB is a triangle.N is a point on AP.
(a) Find the vector , in terms of a and b.
..............................................................................................................................................(1)
B is the midpoint of AC.M is the midpoint of PB.
* (b) Show that NMC is a straight line.
(4)(Total for Question is 5 marks)
Q5.
M is the midpoint of AB.
(a) Write in terms of a and b.
Give your answer in its simplest form.
...........................................................
(2)N is the midpoint of OB.
G is the point on OM such that OG : GM = 2 : 1
*(b) Show that AGN is a straight line.
(4)
(Total for question = 6 marks)
Q6. ∗
OAB is a triangle. M is the midpoint of OA. N is the midpoint of OB.
= m
= n
Show that AB is parallel to MN.
(Total for Question is 3 marks)
Q7.
CAYB is a quadrilateral.
X is the point on AB such that AX : XB = 1 : 2
Prove that
(Total for question is 5 marks)
Q8.
OXYZ is a parallelogram.
P is the point on OX such that OP : PX = 1 : 2 R is the point on OY such that OR : RY = 1 : 3
Work out, in its simplest form, the ratio ZP : ZRYou must show all your working.
...........................................................
(Total for question = 5 marks)
Q9.
OAB is a triangle. A is the midpoint of OZY is the midpoint of ABX is a point on OB
Prove that XYZ is a straight line.
(Total for question = 5 marks)
Q10.
*
OBPA is a quadrilateral.
Y is the point on AP such that AY : YP = 2 : 1
Show that is parallel to the vector 7a + 3b
(Total for question = 4 marks)
Q11.
OPTR is a trapezium.
(a) (i) Find in terms of a and b
...........................................................
(ii) Find in terms of a and bGive your answer in its simplest form.
...........................................................
(2)S is the point on PR such that PS : SR = 1 : 3
(b) Find in terms of a and bGive your answer in its simplest form.
...........................................................
(2)*(c) What does your answer to part (b) tell you about the position of point S?
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(2)
(Total for question = 6 marks)
Q12.
OABC is a parallelogram.
= a and = c
X is the midpoint of the line AC. OCD is a straight line so that OC : CD = k : 1
Given that = 3c − a
find the value of k.
k = ...........................................................
(Total for question = 4 marks)
Q13.
OAN, OMB and APB are straight lines. AN = 2OA. M is the midpoint of OB.
where k is a scalar quantity.
Given that MPN is a straight line, find the value of k.
...........................................................
(Total for question = 5 marks)
Q14.
OMA, ONB and ABC are straight lines. M is the midpoint of OA. B is the midpoint of AC.
where k is a scalar quantity.
Given that MNC is a straight line, find the value of k.
(Total for question = 5 marks)
Q15.
OAB is a triangle. P is the point on AB such that AP : PB = 5:3
= 2a
= 2b
= k(3a + 5b) where k is a scalar quantity.
Find the value of k.
...........................................................
(Total for question = 4 marks)
Q16.
ABC is a straight line.
AB: BC = 2 : 5
Express in terms of a and b. Give your answer in its simplest form.
...........................................................
(Total for question = 4 marks)
Q17.
OAB is a triangle.
M is the midpoint of AB. OMN is a straight line such that ON : OM = 3 : 2
Find, in terms of a and b, an expression for the vector . Write your answer in its simplest form.
...........................................................
(Total for question = 4 marks)
Q18.
The diagram shows a regular hexagon OABCDE.
= a
= b
M is the midpoint of OE. N is the midpoint of AB.
(a) Find in terms of a and/or b.
= ...........................................................
(3)
(b) Describe fully what your answer to part (a) shows about the lines OA and MN.
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(2)
(Total for question = 5 marks)
Q19.
M is the midpoint of BC.Q is the midpoint of AM.
(a) Find in terms of a and b.
= . . . . . . . . . . . . . . . . . .(1)
(b) Find in terms of c.
= . . . . . . . . . . . . . . . . . .(4)
(Total for Question is 5 marks)
Q20.
The diagram is a sketch.
P is the point (2, 4)Q is the point (4, 8)
(a) Find the vector
Give your answer as a column vector
..............................................................................................................................................(2)
=
M is the midpoint of PQ.N is the midpoint of QR.
(b) Find the vector
Give your answer as a column vector
..............................................................................................................................................(3)
(Total for Question is 5 marks)
Q21.
PQRS is a parallelogram.N is the point on SQ such that SN : NQ = 3 : 2
(a) Write down, in terms of a and b, an expression for .
= . . . . . . . . . . . . . . . . . . . . . . (1)
(b) Express in terms of a and b.
= . . . . . . . . . . . . . . . . . . . . . . (3)
(Total for Question is 4 marks)
Q22.
OAB is a triangle.
N is the point on AB such that AN : NB = 3 : 1
(a) Find in terms of a and b.
..............................................................................................................................................(1)
(b) Find in terms of a and b. Give your vector in its simplest form.
..............................................................................................................................................(3)
(Total for Question is 4 marks)
Q23.
Diagram NOT accurately drawn
OAB is a triangle.
= a
= b
(a) Find in terms of a and b. ..............................................................................................................................................
(1)
P is the point on AB such that AP : PB = 3 : 1
(b) Find in terms of a and b.Give your answer in its simplest form.
..............................................................................................................................................(3)
(Total for Question is 4 marks)
Q24.
OAB is a triangle.
(a) Write down the vector in terms of a and b.
...........................................................
(1)
X is the point on AB such that AX : XB = 1 : 4
(b) Express the vector in terms of a and b.
= ...........................................................
(3)
(Total for Question is 4 marks)
Q25.
OACD is a trapezium and OACB is a parallelogram. B is the midpoint of OD. M is the midpoint of CD.
= a and = b
Given that = k × where k is a scalar,
use a vector method to find the value of k.
...........................................................
(Total for question = 3 marks)
Q26.
OACB is a parallelogram.
M is the midpoint of AC.C is the midpoint of the straight line BCX.
= a = b
Prove that OMX is a straight line.
(Total for Question is 4 marks)
Q27.
OAB is a triangle
= 5a
= 2b
T is the point on AB such that AT : TB = 5 : 1
Show that OT is parallel to the vector a + 2b
(Total for question = 4 marks)
Q28.
Diagram NOT accurately drawn
ABCD is a parallelogram.The diagonals of the parallelogram intersect at O.
= a and = b
(a) Write an expression, in terms of a and/or b, for
(i)
. . . . . . . . . . . . . . . . . . . . . .
(ii)
. . . . . . . . . . . . . . . . . . . . . .
(ii)
. . . . . . . . . . . . . . . . . . . . . .
(3)
X is the point such that = 2a – b
(b) (i) Find an expression, in terms of a and b, for
. . . . . . . . . . . . . . . . . . . . .
(ii) B, A and X lie on the same straight line.
Explain why.
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(3)
(Total for Question is 6 marks)
Q29.
(a) On the grid, draw .
(1)
(b) On the grid, mark with a cross the position of D.
Label this point D.(2)
(Total for question = 3 marks)
Q30.
The diagram shows triangle ABC.
D is the point on BC such that BD : DC = 3 : 1
Prove that ADE is a straight line.
(Total for question = 4 marks)
Q31.
OAB is a triangle. OPM and APN are straight lines. M is the midpoint of AB.
OP : PM = 3 : 2
Work out the ratio ON : NB
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(Total for question = 5 marks)
Q32.
CDEF is a quadrilateral.
(a) Express in terms of a and/or b. Give your answer in its simplest form.
...........................................................
(2)M is the midpoint of DE. X is the point on FM such that FX : XM = n : 1 CXE is a straight line.
(b) Work out the value of n.
n = ...........................................................
(4)
(Total for question = 6 marks)
Q33.
(a) Write down as a column vector
(i) a + b
...........................................................
(1)(ii) 2a + 3b
...........................................................
(2)The vector c is drawn on the grid.
(b) From the point P, draw the vector 3c(1)
(Total for question = 4 marks)
Q34.
OACB is a parallelogram.
D is the point such that
The point N divides AB in the ratio 2:1
(a) Write an expression for in terms of a and b.
...........................................................
(3)
*(b) Prove that OND is a straight line.
(3)
(Total for Question is 6 marks)