Quick revision notes · Mathematics 4024 · Chapter 28 of 35
Vectors
Column vectors, magnitude, position vectors, routes in diagrams, ratios, parallel vectors and collinear points.
4 sectionsWorked examples
Common mistakes
At a glance
Column vectors, magnitude, position vectors, routes in diagrams, ratios, parallel vectors and collinear points.
1Column vectors
- A column vector has a top number (x, across) and a bottom number (y, up).
- Add or subtract top with top and bottom with bottom.
- Multiply by a number: multiply both parts. 3 × (−2, 4) = (−6, 12).
- Written a, a (by hand) or AB with an arrow.
Worked example
a = (3, −2) and b = (−1, 5). Find 2a − b.
- 2a = (6, −4).
- (6 − (−1), −4 − 5).
(7, −9)
2Magnitude
- The magnitude |a| is the length of the vector: Pythagoras on its two parts.
Formulas
|(x, y)| = √(x2 + y2)
Worked example
Find the magnitude of (5, −12).
- √(25 + 144) = √169.
13
3Position vectors
- The position vector of A is OA (from the origin).
- AB = OB − OA = b − a.
- Midpoint M of AB has position vector ½(a + b).
Formulas
AB = b − aend minus start
OM = ½(a + b)midpoint
Worked example
A is (2, 3) and B is (7, −1). Find the vector AB.
- B − A = (7 − 2, −1 − 3).
(5, −4)
4Vectors in diagrams
- Find a route along known vectors; going against an arrow changes the sign.
- A point dividing AB in the ratio m : n: AP = m/(m + n) × AB.
- Parallel: one vector is a multiple of the other.
- Collinear A, B, C: AB is a multiple of BC and they share the point B.
How to do it
- Write the route (e.g. O → A → P).
- Replace each step by a vector, sign included.
- Simplify by collecting a's and b's.
Worked example
OA = a, OB = b. P is on AB with AP : PB = 1 : 3. Find OP.
- AB = b − a.
- AP = ¼(b − a).
- OP = a + ¼(b − a).
OP = ¾a + ¼b
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| AB = a − b | Start at A, go back to O, then to B: b − a |
| |(5, −12)| = 5 − 12 | Use Pythagoras: √(25 + 144) = 13 |
| Using ½ for 'ratio 1 : 2' | AP is 1/3 of AB |
| Proving collinear with the multiple only | Also say the vectors share a point |
★In the exam
- Underline vectors in handwriting: a.
- Simplify answers fully: collect the a terms and the b terms.
- For 'show that' questions, write each route step by step.