Key points about vectors
A A vector quantity has both direction and magnitude (size). is a mathematical object that has both Size and a The orientation or angle that a vector is pointing in..
If a car is travelling 30 miles per hour due south, then its velocity has been described with both a magnitude (30 mph) and a direction (south).
Vectors can be represented using either diagrams, column vectors or variables and can be combined and simplified into a The combination of two or more vectors. vector.
Make sure you are familiar with transforming shapes using translation vectors, as this can help when representing vectors.
What is a vector?
A vector describes the movement from one point to another.
Vectors are also used to describe a translation.
A vector can be represented by a line segment labelled with an arrow.
A vector quantity has both direction and Size.
A scalar quantity has only magnitude.
Find out more about vectors below
GCSE exam-style questions
- What column vector is represented by 𝑥 in the diagram?
Use the direction of the arrow to determine the starting point. This vector starts in the bottom right.
The horizontal displacement is 4 squares to the left, so the top value in the column vector is –4.
The vertical displacement is 6 squares up, so the bottom value in the column vector is 6.
- The column vector for \(\overrightarrow{AB}\) is shown in the image.
What is the column vector for \(\overrightarrow{BA}\) ?
The values in column vector \(\overrightarrow{AB}\) mean the displacement from \({A}\) to \({B}\) is 7 squares to the right and 5 squares down.
The displacement from \({B}\) to \({A}\) is the opposite of this, 7 squares to the left and 5 squares up.
Alternatively, the column vector \(\overrightarrow{BA}\) is the negative of \(\overrightarrow{AB}\).
The numbers in the column vector are the same, but the signs are different. The 7 is replace by –7, and the –5 is replaced by 5.
Check your understanding
How to calculate with vectors
It is possible to perform calculations (+, –, ×) with vectors.
The resultant vector can be expressed using a diagram.
To perform calculations, apply the operation to each component of the vector. For example, to add two vectors, add both the top values and the bottom values.
Find out more about vector calculations below
GCSE exam-style questions
- Work out 𝑎 + 𝑏 + 𝑐.
The resultant is found by adding the components in vector 𝑎, 𝑏 and 𝑐.
7 + 1 + (–3) = 5 and –2 + 5 + (–6) = –3.
- Vectors 𝑥 and 𝑦 are drawn on a grid. Using a pencil, ruler and squared paper, draw a vector that represents 𝑥 – 𝑦.
Vector 𝑥 – 𝑦 is a displacement of 4 squares to the right and 2 squares down.
Start by drawing vector 𝑥. Vector 𝑥 is a displacement 4 squares to the right and 3 squares up.
At the end of vector 𝑥, draw vector –𝑦. Vector 𝑦 is a displacement of 5 squares up. So vector –𝑦 is a displacement of 5 squares down.
The resultant vector is shown in the diagram.
How to describe a pathway using vectors
It is possible to describe a pathway around a grid or geometric shape using vectors.
For example, a route from \({A}\) to \({B}\), via a point, \({O}\), can be expressed using vectors:
\(\overrightarrow{AB} = \overrightarrow{𝐴𝑂} + \overrightarrow{𝑂𝐵}\)
When using vectors to describe a route, always use pathways that are already given.
In the triangle in the image to the right, \(\overrightarrow{OA}\) = 𝑎 and \(\overrightarrow{OB}\) = 𝑏.
\(\overrightarrow{AB}\) is equivalent to –𝑎 + 𝑏.
The vectors can be simplified using the same rules as for simplifying algebra expressions.
Find out more below, along with a worked example
GCSE exam-style questions
- The grid shown has been formed by 12 congruent parallelograms.
\(\overrightarrow{OA} = 𝑎\) and \(\overrightarrow{OF} = 𝑓 \).
Express the vector \(\overrightarrow{LJ}\) in terms of 𝑎 and 𝑓.
\(\overrightarrow{LJ} = 3𝑎 - 𝑓\)
Plan a route from \({L}\) to \({J}\) along line segments where the vectors are known.
\(\overrightarrow{LJ} = \overrightarrow{LQ} + \overrightarrow{QJ}\)
The direction of \(\overrightarrow{QJ}\) is in the opposite direction, so a negative vector is used.
\(\overrightarrow{LJ} = 𝑎 + 𝑎 + 𝑎 – 𝑓 \) which simplifies to \(\overrightarrow{LJ} = 3𝑎 – 𝑓 \).
- The grid shown has been formed by 6 congruent equilateral triangles.
\(\overrightarrow{OA} = 𝑎 \) and \(\overrightarrow{OE} = 𝑒 \)
Express the vector \(\overrightarrow{OC}\) in terms of 𝑎 and 𝑒.
\(\overrightarrow{OC} = 2𝑎 + 2𝑒\)
Plan a route from \({𝑂}\) to \({𝐶}\) along line segments where the vectors are known.
\(\overrightarrow{OC} = \overrightarrow{OE} + \overrightarrow{EX} + \overrightarrow{XD} + \overrightarrow{DC} \)
Vectors \(\overrightarrow{EX} \) and \(\overrightarrow{DC} \) are equivalent to 𝑎.
Vector \(\overrightarrow{XD} \) is equivalent to 𝑒.
\(\overrightarrow{OC} = 𝑒 + 𝑎 + 𝑒 + 𝑎 \) which simplifies to \(\overrightarrow{OC} = 2𝑎 + 2𝑒\).
Quiz – Vectors
Practise what you've learned about vectors with this quiz.
Now you've revised vectors, why not look at right-angled trigonometry?
More on Geometry and measure
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