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Maths · Vectors, Constructions and Loci

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Column Vectors and Vector Arithmetic

Writing vectors as columns, adding, subtracting and multiplying them, and spotting parallel vectors.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Write a vector as a column vector, and read one from a diagram.
  2. 2Add and subtract vectors, and multiply a vector by a number.
  3. 3Recognise parallel vectors, and draw vector sums and differences.
  4. 4Find the length of a column vector using Pythagoras' theorem (Higher tier).

Journeys with a size and a direction

A vector describes a movement, with a size and a direction, such as 3 right and 2 up. You have already used vectors to describe translations, and this lesson gives them their own rules. Vectors can be added, subtracted and multiplied by numbers, and the exam asks for each of these both with column vectors and with diagrams. On the non-calculator paper the numbers are small, so the marks go to careful signs and to showing the method, and a vector must always be written as a vector, with an arrow or underline, or as a letter in bold in print.

Writing vectors

A vector can be named by its end points or by a single letter.

  • Column vector

    \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\) means 3 to the right and 2 up. The top number is the horizontal move, and the bottom number is the vertical move.

  • Naming

    \(\overrightarrow{AB}\) is the vector from \(A\) to \(B\). A single letter such as \(\mathbf{a}\) is printed in bold, and written by hand with a line underneath.

  • Reversing

    \(\overrightarrow{BA} = -\overrightarrow{AB}\), so the column vector has both signs changed.

  • Equal vectors

    Two vectors are equal if they have the same size and direction, wherever they are drawn.

Finding a vector from coordinates

The point \(A\) is \((2, 5)\) and the point \(B\) is \((6, 2)\). Write \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) as column vectors.

Show the solutionHide the solution
  1. 1 Horizontal move \(6 - 2 = 4\).
  2. 2 Vertical move \(2 - 5 = -3\), so \(\overrightarrow{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\).
  3. 3 Reverse it \(\overrightarrow{BA} = -\overrightarrow{AB}\), so change both signs.
  4. 4 Check on a sketch From \(A\) to \(B\) goes 4 right and 3 down.

Answer\(\overrightarrow{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\) and \(\overrightarrow{BA} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}\)

Adding, subtracting and multiplying

Do each row on its own, adding or subtracting the top numbers and then the bottom numbers.

  • Adding

    \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\).

  • Subtracting

    \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}\).

  • Multiplying by a number

    \(3 \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\). Both numbers are multiplied.

  • Combining

    \(2\mathbf{a} + 3\mathbf{b}\) is found by working out \(2\mathbf{a}\) and \(3\mathbf{b}\) first, then adding.

Working with column vectors

\(\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\). Work out \(2\mathbf{p} - \mathbf{q}\).

Show the solutionHide the solution
  1. 1 Multiply \(2\mathbf{p} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\).
  2. 2 Subtract the top numbers \(4 - (-1) = 5\).
  3. 3 Subtract the bottom numbers \(6 - 4 = 2\).
  4. 4 Write the answer \(2\mathbf{p} - \mathbf{q} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\).

Answer\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)

Parallel vectors and the length of a vector

Parallel vectors are multiples of each other.

  • Parallel

    \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) are parallel, because the second is 3 times the first. A negative multiple points the other way.

  • Same size

    If \(\mathbf{b} = 2\mathbf{a}\), then \(\mathbf{b}\) is twice as long as \(\mathbf{a}\).

  • Length (Higher tier)

    The length of \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) is \(\sqrt{3^2 + 4^2} = 5\), using Pythagoras.

  • Write the length with bars

    The length of \(\mathbf{a}\) is written \(|\mathbf{a}|\).

Test yourself

  1. 1

    What does \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

    Show answerHide answer

    2 left and 5 up.

  2. 2

    What is \(\overrightarrow{BA}\) if \(\overrightarrow{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\)?

    Show answerHide answer

    \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\).

  3. 3

    What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

    Show answerHide answer

    \(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\).

  4. 4

    Are \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 3 \\ 6 \end{pmatrix}\) parallel?

    Show answerHide answer

    Yes, the second is 3 times the first.

  5. 5

    What is \(3 \begin{pmatrix} 2 \\ -1 \end{pmatrix}\)?

    Show answerHide answer

    \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\).

Exam technique: vectors

Show the working, and keep the signs straight.

  • Write vectors as vectors

    Use an arrow, an underline or the column form, not just letters.

  • Work out the top and bottom rows separately

    It avoids mixing up the signs.

  • Draw it

    A quick sketch with arrows end to end shows whether an answer is sensible.

  • Do not forget the minus

    \(\overrightarrow{BA}\) is not the same as \(\overrightarrow{AB}\).

Summary and exam focus

  • A column vector shows the horizontal move over the vertical move, with right and up positive.
  • Vectors are added and subtracted row by row, and a number multiplies both rows.
  • Parallel vectors are multiples of each other.
  • The length of \(\begin{pmatrix} x \\ y \end{pmatrix}\) is \(\sqrt{x^2 + y^2}\) (Higher tier).

Exam focus

\(\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\). Work out \(\mathbf{a} + 2\mathbf{b}\). (2 marks) (2 marks)

First find \(2\mathbf{b} = \begin{pmatrix} 2 \\ 8 \end{pmatrix}\), then add: \(\begin{pmatrix} 3 + 2 \\ -2 + 8 \end{pmatrix} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}\). Showing \(2\mathbf{b}\) earns the method mark.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Vector
A quantity with a size and a direction.
Column vector
A vector written with the horizontal move above the vertical move.
Scalar
An ordinary number, such as 3, used to multiply a vector.
Parallel
Pointing in the same or opposite direction, with one vector a multiple of the other.
Resultant
The single vector that has the same effect as a combination of vectors.
Magnitude
The length of a vector.
Displacement
A change in position, with a direction.
Opposite vector
A vector of the same length in the opposite direction.
Triangle rule
Vectors added end to end give a resultant from the start to the finish.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 2 marks Easier

Work out \(\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ 4 \end{pmatrix}\). [2 marks]

Mark scheme — 2 marks available

  • Adds the top numbers or the bottom numbers — M1
  • \(\begin{pmatrix} 5 \\ 5 \end{pmatrix}\) — A1

Model answer

\(\begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}\).

2. Exam question Work out 4 marks Core

The vectors \(\mathbf{r}\) and \(\mathbf{s}\) are drawn on the grid. (a) Write \(\mathbf{r}\) and \(\mathbf{s}\) as column vectors. [2 marks] (b) Work out \(3\mathbf{r} - \mathbf{s}\). [2 marks]

Two vectors r and s drawn as arrows on a grid.

Mark scheme — 4 marks available

  • (a) \(\mathbf{r} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\) — B1
  • (a) \(\mathbf{s} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\) — B1
  • (b) \(3\mathbf{r} = \begin{pmatrix} 6 \\ 12 \end{pmatrix}\) or a correct method — M1
  • (b) \(\begin{pmatrix} 3 \\ 13 \end{pmatrix}\) — A1 (follow through from (a))

Model answer

(a) \(\mathbf{r}\) goes 2 right and 4 up, so \(\mathbf{r} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\). \(\mathbf{s}\) goes 3 right and 1 down, so \(\mathbf{s} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\). (b) \(3\mathbf{r} = \begin{pmatrix} 6 \\ 12 \end{pmatrix}\), so \(3\mathbf{r} - \mathbf{s} = \begin{pmatrix} 3 \\ 13 \end{pmatrix}\).

3. Exam question Work out 3 marks Core

\(P\) is the point \((2, -1)\) and \(Q\) is the point \((-3, 4)\). (a) Write \(\overrightarrow{PQ}\) as a column vector. [2 marks] (b) Write \(\overrightarrow{QP}\) as a column vector. [1 mark]

Mark scheme — 3 marks available

  • (a) \(-3 - 2\) or \(4 - (-1)\) — M1
  • (a) \(\begin{pmatrix} -5 \\ 5 \end{pmatrix}\) — A1
  • (b) \(\begin{pmatrix} 5 \\ -5 \end{pmatrix}\) — B1 (follow through from (a))

Model answer

(a) \((-3 - 2, 4 - (-1)) = (-5, 5)\), so \(\overrightarrow{PQ} = \begin{pmatrix} -5 \\ 5 \end{pmatrix}\). (b) \(\overrightarrow{QP} = \begin{pmatrix} 5 \\ -5 \end{pmatrix}\).

4. Exam question Work out 3 marks Core

\(\mathbf{a} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}\). (a) Work out \(2\mathbf{a} - \mathbf{b}\). [2 marks] (b) Work out \(\mathbf{a} + \mathbf{b}\). [1 mark]

Mark scheme — 3 marks available

  • (a) \(2\mathbf{a} = \begin{pmatrix} -4 \\ 10 \end{pmatrix}\) or a correct method — M1
  • (a) \(\begin{pmatrix} -8 \\ 9 \end{pmatrix}\) — A1
  • (b) \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\) — B1

Model answer

(a) \(2\mathbf{a} = \begin{pmatrix} -4 \\ 10 \end{pmatrix}\), so \(2\mathbf{a} - \mathbf{b} = \begin{pmatrix} -8 \\ 9 \end{pmatrix}\). (b) \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\).

5. Exam question Write down 2 marks Core

Write down a vector that is parallel to \(\begin{pmatrix} 1 \\ -3 \end{pmatrix}\) and four times as long. [2 marks]

Mark scheme — 2 marks available

  • Multiplies both numbers by 4 — M1
  • \(\begin{pmatrix} 4 \\ -12 \end{pmatrix}\) — A1

Model answer

Multiply by 4: \(4 \times \begin{pmatrix} 1 \\ -3 \end{pmatrix} = \begin{pmatrix} 4 \\ -12 \end{pmatrix}\).

6. Exam question Show that 3 marks Stretch

\(\overrightarrow{AB} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\overrightarrow{BC} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\). Show that \(A\), \(B\) and \(C\) lie on a straight line. [3 marks]

Mark scheme — 3 marks available

  • \(\overrightarrow{BC} = 2\overrightarrow{AB}\) — M1
  • Parallel — A1
  • Common point \(B\), so collinear — Q1

Model answer

\(\overrightarrow{BC} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} = 2 \times \begin{pmatrix} 2 \\ 3 \end{pmatrix} = 2\overrightarrow{AB}\). The vectors are parallel and share the point \(B\), so \(A\), \(B\) and \(C\) are on a straight line.

7. Multiple choice 1 mark Easier

What does the column vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

  1. A 2 right and 5 up
  2. B 2 left and 5 up Correct
  3. C 5 left and 2 up
  4. D 2 left and 5 down

Why: The top number is the horizontal move, and the bottom number is the vertical move.

8. Multiple choice 1 mark Core

\(A\) is \((2, 5)\) and \(B\) is \((6, 2)\). What is \(\overrightarrow{AB}\)?

  1. A \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) Correct
  2. B \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\)
  3. C \(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\)
  4. D \(\begin{pmatrix} 8 \\ 7 \end{pmatrix}\)

Why: Subtract the start from the end: \((6 - 2, 2 - 5) = (4, -3)\).

9. Multiple choice 1 mark Easier

What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

  1. A \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)
  2. B \(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\)
  3. C \(\begin{pmatrix} 10 \\ 10 \end{pmatrix}\)
  4. D \(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\) Correct

Why: Add the top numbers and add the bottom numbers.

10. Multiple choice 1 mark Easier

What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

  1. A \(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\)
  2. B \(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\)
  3. C \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\) Correct
  4. D \(\begin{pmatrix} 2 \\ 2 \end{pmatrix}\)

Why: \((3 - 1, 2 - 4) = (2, -2)\).

11. Multiple choice 1 mark Easier

What is \(3\begin{pmatrix} 2 \\ -1 \end{pmatrix}\)?

  1. A \(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
  2. B \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\) Correct
  3. C \(\begin{pmatrix} 6 \\ -1 \end{pmatrix}\)
  4. D \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\)

Why: Multiply both numbers by 3.

12. Multiple choice 1 mark Core

\(\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\). What is \(2\mathbf{p} - \mathbf{q}\)?

  1. A \(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\) Correct
  2. B \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
  3. C \(\begin{pmatrix} 3 \\ 10 \end{pmatrix}\)
  4. D \(\begin{pmatrix} 5 \\ 10 \end{pmatrix}\)

Why: \(2\mathbf{p} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\), then \((4 - (-1), 6 - 4) = (5, 2)\).

13. Multiple choice 1 mark Core

Which vector is parallel to \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)?

  1. A \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
  2. B \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\)
  3. C \(\begin{pmatrix} 4 \\ 5 \end{pmatrix}\)
  4. D \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) Correct

Why: \(\begin{pmatrix} 6 \\ 9 \end{pmatrix} = 3\begin{pmatrix} 2 \\ 3 \end{pmatrix}\), so it is parallel.

14. Multiple choice 1 mark Easier

\(\overrightarrow{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\). What is \(\overrightarrow{BA}\)?

  1. A \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\)
  2. B \(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\)
  3. C \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\) Correct
  4. D \(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\)

Why: \(\overrightarrow{BA} = -\overrightarrow{AB}\), so both signs change.

15. Multiple choice 1 mark Core

What is the length of the vector \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)?

  1. A \(7\)
  2. B \(5\) Correct
  3. C \(25\)
  4. D \(12\)

Why: The length is \(\sqrt{3^2 + 4^2} = \sqrt{25} = 5\).