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

Vectors in three dimensions

Column and i j k notation, magnitude, distance between points, position vectors, parallel and collinear points, geometry problems in 3D.

12 original questions, 64 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

The vector a = 2i − 3j + 6k

  1. Find |a|
  2. Find a unit vector in the direction of a
  3. Find the angle that a makes with the positive x-axis, giving your answer in degrees to 1 decimal place.
4 marks
Show answer
a) 7 b) 17(2i - 3j + 6k) c) 73.4°
2

Relative to a fixed origin O, the point A has coordinates (3, −1, 4) and the point B has coordinates (−1, 2, 16).

  1. Find →AB in the form pi + qj + rk
  2. Find the distance AB.
  3. Find the coordinates of the midpoint of AB.
3 marks
Show answer
a) -4i + 3j + 12k b) 13 c) (1, 12, 10)
3

The vectors p = 3i + aj − 6k and q = bi − 4j + 8k, where a and b are constants, are parallel.

Find the value of a and the value of b.

3 marks
Show answer
a = 3, b = -4
4

Relative to a fixed origin O, the point A has position vector 4i − 2j + k and the point B has position vector pi + 3j − k, where p is a positive constant.

Given that |→AB| = 7, find the exact value of p.

4 marks
Show answer
p = 4 + 2√5
5

Relative to a fixed origin O, the points A, B and C have coordinates (1, 2, −3), (4, 0, 1) and (10, −4, 9) respectively.

  1. Show that A, B and C are collinear.
  2. Find the ratio AB : BC
4 marks
Show answer
a) shown: →BC = 2→AB and B is common b) 1 : 2
6

The triangle ABC has

→AB = 2i − j + 2k   and   →BC = 4i + 4j − 2k

  1. Find →AC
  2. Show that angle ABC is a right angle.
  3. Find the area of triangle ABC.
5 marks
Show answer
a) 6i + 3j b) shown: AB2 + BC2 = 9 + 36 = 45 = AC2 c) 9
7

Relative to a fixed origin O, the points P, Q and R have position vectors

p = i + 2k,   q = 3i + 4j − 2k,   r = 5j + k

  1. Find the size of angle QPR, giving your answer in degrees to 1 decimal place.
  2. Find the area of triangle PQR, giving your answer to 3 significant figures.
6 marks
Show answer
a) 45.1° b) 11.0
8

Relative to a fixed origin O, the point A has position vector 2i − j + 5k and the point B has position vector 8i + 5j − 4k

The straight line l passes through A and B.

  1. Find →AB
  2. Given that the point P lies on l such that |→AP| = 2|→BP|, find the two possible position vectors of P.
5 marks
Show answer
a) 6i + 6j - 9k b) 6i + 3j - k or 14i + 11j - 13k
9

Relative to a fixed origin O,

• the point A has position vector 2i + 3j − k
• the point B has position vector 7i + 4j + 3k
• the point C has position vector 3i − 2j + 5k

  1. Find →AB and →CB
  2. Show that the quadrilateral OABC is a trapezium, giving reasons for your answer.
  3. Find the ratio OA : CB
5 marks
Show answer
a) →AB = 5i + j + 4k, →CB = 4i + 6j - 2k b) shown: CB = 2OA so OA ∥ CB, but AB and OC are not parallel c) 1 : 2
10

Relative to a fixed origin O, the points A, B and C have coordinates (2, 4, −1), (5, 3, 4) and (3, −1, 5) respectively.

  1. Show that OABC is a parallelogram.
  2. Find the size of angle AOC, giving your answer in degrees to 1 decimal place.
  3. Find the area of the parallelogram OABC, giving your answer to 3 significant figures.
8 marks
Show answer
a) shown: →OA = →CB and →OC = →AB b) 96.4° c) 26.9
11

An adventure park builds a zip wire. Relative to a fixed origin O at the base of the starting tower, the unit vectors i, j and k point east, north and vertically upwards, and distances are in metres.

The wire is modelled as a straight line from the top of the tower, T, with position vector 12k, to the landing point L, with position vector 40i + 30j + 2k

  1. Find the length of the wire.
  2. Find the angle the wire makes with the horizontal, giving your answer in degrees to 1 decimal place.
  3. A rider is three quarters of the way along the wire from T to L. Find the height of the rider above the ground, which is horizontal.
  4. Give one reason why the model of the wire as a straight line may not be realistic.
8 marks
Show answer
a) 51.0 m b) 11.3° c) 4.5 m d) A real wire sags under its own weight and the rider's weight, so it is not straight
12

Relative to a fixed origin O, the points A, B and C have coordinates (2, 0, 1), (4, 3, −1) and (a, 5, a) respectively, where a is a constant.

Given that |→AB| = |→BC|,

  1. show that a2 − 3a + 2 = 0 and hence find the two possible values of a.
  2. For the smaller value of a, find the coordinates of the point D such that ABCD is a parallelogram, and explain why ABCD is a rhombus.
  3. For the smaller value of a, find the size of angle ABC, giving your answer in degrees to 1 decimal place.
9 marks
Show answer
a) a = 1 or a = 2 b) D(-1, 2, 3); a parallelogram with two adjacent sides equal (AB = BC) is a rhombus c) 76.4°

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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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