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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Vectors

Column and i j notation, magnitude and direction, unit vectors, position vectors, geometry with vectors, parallel vectors.

12 original questions, 63 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

The vectors a and b are given by a = 3−4 and b = −25

  1. Find 2a − 3b, giving your answer in terms of i and j
  2. Find |a|
3 marks
Show answer
a) 12i - 23j b) |a| = 5
2

The vector v = 5i − 12j

  1. Find a unit vector in the direction of v
  2. Find the angle between v and the positive i direction.
4 marks
Show answer
a) 513i - 1213j b) 67.4° (below the i direction)
3

Relative to a fixed origin O, the point A has position vector 2i − j and the point B has position vector −4i + 7j

  1. Find →AB
  2. Find the distance AB
  3. Find the position vector of the midpoint of AB
4 marks
Show answer
a) -6i + 8j b) 10 c) -i + 3j
4

The vectors p = (k + 2)i + 6j and q = 2i + (k + 3)j, where k is a constant.

Given that p is parallel to q, find the possible values of k.

4 marks
Show answer
k = 1 or k = -6
5

The vector c = 2ti + (t − 5)j, where t is a constant.

Given that |c| = √65

  1. find the possible values of t
  2. find, for the positive value of t, a unit vector in the direction of c
5 marks
Show answer
a) t = 4 or t = -2 b) 1√65(8i - j)
6

Relative to a fixed origin O, the points A and B have position vectors

→OA = 3i + j    and    →OB = −i + 4j

  1. Find the size of angle AOB, giving your answer in degrees to one decimal place.
  2. Find the exact area of triangle OAB
6 marks
Show answer
a) 85.6° b) 132
7

Relative to a fixed origin O, the points A, B, C and D have position vectors

a = i + 2j,   b = 5i + 4j,   c = 8i + 10j,   d = 2i + 7j

  1. Show that ABCD is a trapezium, giving reasons for your answer.
  2. Find the ratio AB : DC
4 marks
Show answer
a) DC = 32AB so AB ∥ DC, and AD is not parallel to BC, so ABCD is a trapezium b) 2 : 3
8

Relative to a fixed origin O, the point A has position vector 4i − 7j and the point B has position vector 14i + 8j

The point P lies on the line segment AB such that AP : PB = 3 : 2

  1. Find the position vector of P
  2. Find |→OP|, giving your answer as a simplified surd.
4 marks
Show answer
a) 10i + 2j b) 2√26
9

In triangle OAB, →OA = a and →OB = b

The point M is the midpoint of OA. The point N lies on AB such that AN : NB = 2 : 1

  1. Find →MN in terms of a and b, giving your answer in its simplest form.

The point P is such that →OP = 2b

  1. Show that M, N and P lie on a straight line.
  2. Find the ratio MN : NP
6 marks
Show answer
a) MN = -16a + 23b b) MP = 3MN, a multiple with the common point M, so M, N and P are collinear c) 1 : 2
10

A park is modelled as a quadrilateral OPQR. Relative to the point O, the corners have position vectors

→OP = 6i,   →OQ = 10i + 9j,   →OR = 8i + 18j

where i and j are unit vectors due east and due north and each unit represents 10 metres.

  1. Show that the side PQ is parallel to the side OR
  2. A straight path runs from O to Q. Find the length of the path in metres.
  3. Find the size of angle OPQ
  4. Give one limitation of the model.
8 marks
Show answer
a) PQ = 4i + 9j and OR = 2PQ, so they are parallel b) 135 m (3 s.f.) c) 114.0° d) e.g. the edges of the park will not be exactly straight lines, or the ground is not flat
11

Relative to a fixed origin O, the points A, B and C have position vectors

i + 2j,   4i + 6j   and   pi − 3j

respectively, where p is a constant. Given that angle ABC = 90°

  1. use Pythagoras' theorem to find the value of p
  2. find the area of triangle ABC
  3. The point D is such that ABCD is a rectangle. Find the position vector of D
7 marks
Show answer
a) p = 16 b) 37.5 c) 13i - 7j
12

In triangle OAB, →OA = a and →OB = b

The point P lies on OA such that OP : PA = 1 : 2 and the point Q is the midpoint of AB. The lines OQ and BP meet at the point X.

  1. Find, in terms of a and b, (i) →OQ   (ii) →BP
  2. Using a vector method, find the ratio OX : XQ
  3. Find the ratio BX : XP
8 marks
Show answer
a) OQ = 12a + 12b; BP = 13a - b b) 1 : 1 c) 3 : 1

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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