Vectors
A vector has a size and a direction. At GCSE you write it as a column vector, add and scale it, and on the Higher tier use vectors to show that lines are parallel or that points lie on a straight line.
Part 1 of 3: Learn it
In short
- In a column vector, the top number is the move right (or left if negative), the bottom is up (or down).
- Add vectors by adding the tops and adding the bottoms.
- The vector from A to B is b − a. If one vector is a number times another, the two are parallel.
Where this is in your specification
Spec points: DfE G24 and G25 (vector geometry proofs are Higher tier)
| Board | Topic: Vectors |
|---|---|
| DfE content | G24, G25 |
| AQA 8300 | Geometry and measures |
| Edexcel 1MA1 | Geometry and measures |
| OCR J560 | 11 Vectors |
| Eduqas C300 | Geometry and measures: vectors (end of the geometry content) |
| Cambridge IGCSE 0580 | E7.2 to E7.4 |
| National 5 C847 75 | Geometric skills: vectors in two and three dimensions |
Column vectors
The vector (3 over −2) means 3 right and 2 down. Written as a bold letter, such as a, or with an arrow over two letters, such as AB, which goes from A to B. The vector BA is the same length in the opposite direction, so BA = −AB.
Vector arithmetic
| Operation | Rule | Example with a = (3 over −1), b = (−2 over 4) |
|---|---|---|
| Add | add tops, add bottoms | a + b = (1 over 3) |
| Subtract | subtract tops, subtract bottoms | a − b = (5 over −5) |
| Scale | multiply both numbers | 3a = (9 over −3) |
Multiplying by a negative number reverses the direction. Multiplying by 2 doubles the length and keeps the direction.
Vectors between points
From A(1, 2) to B(7, −1), subtract the coordinates of A from those of B: AB = (6 over −3). In a diagram, to get from one point to another, follow any route made of known vectors and add them, using a minus sign for any vector you travel against.
Vector geometry (Higher)
- The midpoint M of AB, from the origin: OM = a + ½(b − a) = ½a + ½b.
- Two vectors are parallel if one is a number times the other: 2p + 4q = 2(p + 2q), so it is parallel to p + 2q.
- Three points are on a straight line if two of the vectors joining them are parallel and share a point.
Work out (5 over 2) − (1 over −3).
Show the answer
(4 over 5).
Part 2 of 3: See it worked
Worked examples
Example 1
a = (3 over −1) and b = (−2 over 4). Find 2a + b and a − 3b.
- 2a = (6 over −2), so 2a + b = (6 − 2 over −2 + 4) = (4 over 2)
- 3b = (−6 over 12)
- a − 3b = (3 + 6 over −1 − 12)
Answer: 2a + b = (4 over 2) and a − 3b = (9 over −13).
Example 2
OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b. (Higher)
- AB = −a + b = b − a
- AM = ½(b − a)
- OM = OA + AM = a + ½b − ½a
Answer: OM = ½a + ½b, which is ½(a + b).
Common mistakes
- Writing AB as a − b. From A to B is b − a.
- Adding a top number to a bottom number.
- Forgetting to multiply both numbers when scaling a column vector.
- Ending a proof with the algebra but no sentence saying the lines are parallel.
Work out 3 × (−2 over 1).
Show the answer
(−6 over 3).
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Work out (5 over 2) − (1 over −3).
(4 over 5).
Work out 3 × (−2 over 1).
(−6 over 3).
Are (2 over −3) and (−6 over 9) parallel?
Yes: (−6 over 9) = −3 × (2 over −3), so they are parallel, pointing in opposite directions.
Jobs that use this
- Aerospace engineer (opens a new tab)
- Computer games developer (opens a new tab)
- Airline pilot (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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