Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Vectors. unlimited generated questions on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
A vector has size and direction. You add, subtract and multiply column vectors, draw and use them on a grid, and on Higher write routes through a diagram in terms of vectors such as a and b to prove that lines are parallel or that points lie on a straight line.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
4
Read and draw column vectorsThe top number is the move across and the bottom number the move up or down.
4
Add and subtract column vectorsAdd or subtract the top numbers and the bottom numbers separately.
5
Multiply by a scalar and combine vectorsMultiply both numbers by the scalar, e.g. work out 2a − 3b as a column vector.
5
Write a vector path using a and bGo along known vectors, e.g. \(\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}\).
Notes
Vector basics
A vector has size (magnitude) and direction. A scalar is just a number.
The column vector \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\) means 3 right and 2 down.
Vectors are written \(\overrightarrow{AB}\) (from A to B) or with a bold letter such as \(\mathbf{a}\). When you write by hand, underline the letter.
\(\overrightarrow{BA} = -\overrightarrow{AB}\): the same size in the opposite direction.
Calculating with vectors
Add or subtract the top numbers and the bottom numbers separately: \(\begin{pmatrix} 3 \\ -2 \end{pmatrix} + \begin{pmatrix} 1 \\ 5 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}\).
Multiplying by a scalar multiplies both numbers: \(3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\). The result is parallel to the original and 3 times as long.
On a diagram, a + b means follow a, then follow b from where a ends. The result (the resultant) goes straight from the start to the end.
Vector paths
To find a vector such as \(\overrightarrow{AB}\), go from A to B along a route of known vectors. Going against an arrow makes that vector negative.
Multiply each vector by its scalar (both numbers).
Add or subtract the top numbers, then the bottom numbers.
Write the answer as a column vector.
Example. \(\mathbf{a} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}\). Work out \(2\mathbf{a} - 3\mathbf{b}\) as a column vector.
Choose a route from the start point to the end point along vectors you know.
Use a minus sign for any part of the route that goes against an arrow.
Simplify: collect the a terms and the b terms.
Example. OACB is a parallelogram. \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). M is the midpoint of AC. Write in terms of a and b: (a) \(\overrightarrow{AB}\) (b) \(\overrightarrow{OC}\) (c) \(\overrightarrow{OM}\)
Going the wrong way along an arrow? Put a minus sign in front.
End minus start: if \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\), then \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\).
Higher: a ratio of 1 : 3 splits the line into 1 + 3 = 4 parts, so AP is ¼ of AB.
Higher: collinear checklist: (1) a multiple, (2) a common point, (3) a concluding sentence.