Robinson Tuition
Grade 8 · Higher · free GCSE maths worksheet

Vector proofs

Showing lines are parallel or points are collinear.

13 original questions, 58 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

OAB is a triangle.

Vector OA = a and vector OB = b.

M is the midpoint of OA and N is the midpoint of OB.

Prove that MN is parallel to AB.

abOABMNDiagram NOT accurately drawn
4 marks
Show answer
MN = 12(b - a) = 12AB, so MN is a scalar multiple of AB and the lines are parallel
2

ABCD is a quadrilateral.

Vector AB = 4a + 2b
Vector BC = a − 3b
Vector CD = −4a − 2b

Prove that ABCD is a parallelogram.

4 marks
Show answer
DC = -CD = 4a + 2b = AB, so AB and DC are equal and parallel, which proves ABCD is a parallelogram
3

Vector EF = 4a − 2b

Vector GH = 6a − 3b

Prove that EF is parallel to GH.

4 marks
Show answer
GH = 32EF, so GH is a scalar multiple of EF and the lines are parallel
4

P, Q and R are three points.

Vector PQ = 2a − 3b

Vector QR = 6a − 9b

Prove that P, Q and R lie on the same straight line.

4 marks
Show answer
QR = 3PQ, so PQ and QR are parallel; they share the point Q, so P, Q and R are collinear
5

OAB is a triangle.

Vector OA = 2a and vector OB = 3b.

M is the midpoint of OA.

N is the point on OB such that ON : NB = 2 : 1.

  1. Find vector MN in terms of a and b.
  2. Show that MN is not parallel to AB.
2a3bOABMNDiagram NOT accurately drawn
5 marks
Show answer
a) MN = 2b - a b) If 2b - a = k(3b - 2a) then k = 1/2 from the a terms and k = 2/3 from the b terms; these are different, so MN is not parallel to AB
6

OACB is a parallelogram.

Vector OA = a and vector OB = b, so vector OC = a + b.

M is the midpoint of OB.
X is the midpoint of OC.
D is the midpoint of AC.

Prove that M, X and D lie on the same straight line.

abOABCMXDDiagram NOT accurately drawn
5 marks
Show answer
MX = 12a and XD = 12a, so MX = XD; the segments are parallel and share the point X, so M, X and D are collinear
7

The points A and B have position vectors

OA = 4a + 3b and OB = 8a + 11b.

  1. Find vector AB in terms of a and b.
  2. The point P lies on AB such that AP : PB = 3 : 1. Work out vector OP in terms of a and b.
  3. The point D has position vector OD = 14a + 18b. Prove that O, P and D lie on the same straight line.
5 marks
Show answer
a) AB = 4a + 8b b) OP = 7a + 9b c) OD = 14a + 18b = 2(7a + 9b) = 2OP, so OD is a scalar multiple of OP; both pass through O, so O, P and D are collinear
8

The vectors a and b are non-parallel.

The points A, B and C have position vectors

OA = 2a + 5b
OB = 4a + λb
OC = 8a + 23b

where λ is a number.

A, B and C lie on the same straight line.

Find the value of λ.

5 marks
Show answer
11
9

OAB is a triangle.

Vector OA = a and vector OB = b.

M is the midpoint of AB.
N is the midpoint of OB.
P is the point on OM such that OP : PM = 2 : 1.

  1. Prove that A, P and N lie on the same straight line.
  2. Find AP : PN.
abOABMNPDiagram NOT accurately drawn
5 marks
Show answer
a) AP = -(2/3)a + (1/3)b and AN = -a + (1/2)b = (3/2)AP, so AN is a scalar multiple of AP; both pass through A, so A, P and N are collinear b) 2 : 1
10

OAB is a triangle.

Vector OA = 3a and vector OB = 3b.

P is the point on AB such that AP : PB = 1 : 2.

  1. Find vector AB in terms of a and b.
  2. Show that vector OP = 2a + b.
  3. The point Q lies on OP produced, so that BQ is parallel to OA. Work out OP : PQ.
3a3bOABPQDiagram NOT accurately drawn
6 marks
Show answer
a) AB = 3b - 3a b) OP = OA + (1/3)AB = 3a + b - a = 2a + b c) 1 : 2
11

OAB is a triangle. OA = a and OB = b.

M is the midpoint of OA and N is the midpoint of OB.

Show that MN is parallel to AB.

abOABMNDiagram NOT accurately drawn
3 marks
Show answer
MN = (12)(b - a) = (12)AB, so MN is parallel to AB
12

OAB is a triangle. OA = a and OB = b.

P is the point on AB such that AP : PB = 2 : 1.

Express OP in terms of a and b. Give your answer in its simplest form.

abOABPDiagram NOT accurately drawn
4 marks
Show answer
(13)a + (23)b
13

OABC is a trapezium. OA = a and OC = c.

CB is parallel to OA and CB = 2a.

M is the midpoint of AB.

Find OM in terms of a and c. Give your answer in its simplest form.

ac2aOACBMDiagram NOT accurately drawn
4 marks
Show answer
(32)a + (12)c

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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