A vector has size and direction, and can be written as a column vector of horizontal and vertical movement. Add or subtract column vectors by combining the top numbers and the bottom numbers separately, and multiply by a scalar by multiplying both. The magnitude of a vector is its length, found with Pythagoras. In vector geometry, routes between points are combinations of given vectors, and parallel vectors are multiples of each other.
Vector rules
- Add or subtract: combine the top numbers, then the bottom numbers
- Scalar multiple: multiply both components
- Magnitude of (x, y): √(x² + y²)
- AB = b − a, where a and b are the position vectors of A and B
- Parallel vectors: one is a multiple of the other
Column vectors
A column vector has a top number for the horizontal movement and a bottom number for the vertical movement. Adding vectors is like making one journey after another, so a + b means follow a and then b. Add the top numbers together and the bottom numbers together. Subtracting works the same way: a − b means a + (−b).
Scalar multiples and magnitude
Multiplying a vector by a number, a scalar, multiplies both components. It makes the vector longer or shorter in the same direction, or reverses it if the scalar is negative. The magnitude, written |a|, is the length of the vector, found by Pythagoras: the magnitude of (7, 24) is √(49 + 576) = 25.
Routes in vector geometry
In geometry questions, vectors are usually given as letters, such as OA = a and OB = b. To get from A to B, go back along a and then out along b, so AB = −a + b = b − a. Any route between two points gives the same vector, so choose the route that uses vectors you know.
Midpoints and ratios
The midpoint M of AB is halfway along, so AM = ½AB. From O, OM = OA + AM = a + ½(b − a) = ½(a + b). If a point divides a line in a ratio such as 1 : 2, it is one third of the way along.
Proving lines are parallel
Two vectors are parallel if one is a scalar multiple of the other. For example, 2a + 4b = 2(a + 2b), so it is parallel to a + 2b and twice as long. If two parallel vectors share a point, the points lie on a straight line, which proves they are collinear.
Worked examples
Example 1. a = (2, 5) and b = (3, −4). Work out a + b and a − b.
- a + b: 2 + 3 = 5 and 5 + (−4) = 1.
- a − b: 2 − 3 = −1 and 5 − (−4) = 9.
Answer: a + b = (5, 1); a − b = (−1, 9).
Example 2. Using the same vectors, work out 3a − 2b.
- 3a = (6, 15) and 2b = (6, −8).
- 3a − 2b = (6 − 6, 15 − (−8)).
Answer: (0, 23).
Example 3. Find the magnitude of the vector (7, 24).
- √(7² + 24²) = √(49 + 576) = √625.
Answer: 25.
Example 4. OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.
- AB = b − a, so AM = ½(b − a).
- OM = a + ½(b − a).
Answer: OM = ½(a + b).
Example 5. PQ = 2a + 4b and RS = a + 2b. Show that PQ and RS are parallel.
- PQ = 2a + 4b = 2(a + 2b) = 2RS.
Answer: PQ is a multiple of RS, so they are parallel, and PQ is twice as long.
Common mistakes
- Adding top and bottom numbers together. Combine top with top and bottom with bottom.
- Sign errors when subtracting negatives. 5 − (−4) = 9.
- Getting AB the wrong way round. AB = b − a, not a − b.
- Forgetting to square-root for magnitude. |a| = √(x² + y²).
- Not simplifying to show parallel vectors. Factorise to show one vector is a multiple of the other.
How vectors are examined
Column vector arithmetic can come up on Foundation or Higher papers, while vector geometry proofs are reserved for Higher. Questions ask you to:
- add, subtract and multiply column vectors
- find the magnitude of a vector
- write a route between points in terms of given vectors
- prove that lines are parallel or that points are collinear
The vectors worksheet includes answers, and the GCSE geometry revision engine generates more questions.
Column vectors are used for translations, and vectors continue at A-Level in vector equations of lines.
Frequently asked questions
How do you add column vectors?
Add the top numbers together and the bottom numbers together.
How do you find the magnitude of a vector?
Use Pythagoras: the magnitude of (x, y) is √(x² + y²).
How do you show two vectors are parallel?
Show that one is a number multiplied by the other.