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Column Vectors

FoundationHigherAQAEdexcelOCR

Get to grips with column vectors using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on vector arithmetic and geometric vector proofs, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. The top number moves right/left; the bottom number moves up/down.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A column vector describes a movement using two numbers written vertically. The top number gives the horizontal movement and the bottom number the vertical movement.

So \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) means four right and three down. Positive is right and up; negative is left and down.

Vectors add by adding the top numbers together and the bottom numbers together, keeping them separate. Multiplying by a number scales both components, so doubling a vector doubles both the horizontal and vertical movements.

Revision notes

Reading a column vector

The top number is horizontal, the bottom vertical. Positive means right or up.

So \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) means two left and five up.

Adding and subtracting

Add the top numbers together and the bottom numbers together, treating each row separately.

\(\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}\).

Multiplying by a scalar

Multiply both components by the number.

So \(3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\). The direction stays the same but the length changes.

Key points

  • The top number is horizontal movement.
  • The bottom number is vertical movement.
  • Positive is right or up.
  • Add vectors row by row.
  • Multiplying by a scalar scales both components.
  • Write vectors vertically in brackets.

Worked examples

Example 1

Work out \(\begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix}\).

Working

\[5 + (-1) = 4\]add the top numbers
\[2 + 3 = 5\]add the bottom numbers
\[\begin{pmatrix} 4 \\ 5 \end{pmatrix}\]write the resulting vector

Example 2

Work out \(4\begin{pmatrix} 3 \\ -2 \end{pmatrix}\).

Working

\[4 \times 3 = 12\]multiply the top number
\[4 \times -2 = -8\]multiply the bottom number
\[\begin{pmatrix} 12 \\ -8 \end{pmatrix}\]write the resulting vector

Example 3

Describe the movement given by \(\begin{pmatrix} -3 \\ -4 \end{pmatrix}\).

Working

\[\text{Negative top means left}\]interpret the horizontal component
\[\text{3 left and 4 down}\]interpret both components

Common mistakes

  • Reversing the components.

    The top number is always horizontal, the bottom vertical.

  • Adding across rather than down.

    Top adds to top and bottom to bottom; never mix them.

  • Multiplying only one component by a scalar.

    Both numbers must be multiplied.

  • Writing a vector as a coordinate.

    Column vectors are written vertically in brackets, not as (x, y).

Exam tips

  • Keep the two rows separate in every calculation.
  • Write vectors vertically rather than as coordinates.
  • Check the signs match the direction described.
  • Multiply both components when scaling.

Key terms

Column vector
Two numbers written vertically describing a movement.
Scalar
A number that multiplies a vector.
Component
One of the two numbers in a vector.
Magnitude
The length of a vector.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.