The choice between coordinates and vectors is made from the wording, and the reason takes one sentence. Coordinates suit equations and areas, while vectors suit ratios, parallel lines and letters in place of numbers.
This lesson opens geometry problems with several possible approaches, part of coordinate geometry.
Which wording points to which method?
| The question asks for | Method | Why |
|---|---|---|
| Equation of a line, gradient, area | Coordinates | These come straight from the numbers |
| Point dividing a segment in a ratio | Either | Section formula or position vectors |
| Show that points are collinear or lines parallel | Vectors | One vector is a multiple of another |
| Given OA = a and OB = b | Vectors | The data is in letters, not numbers |
Worked example: the same problem, two ways
The points are A(2, 1) and B(8, 7). Part (a): find the point P on AB with AP : PB = 1 : 2. Part (b): find the equation of the line through P perpendicular to AB.
By coordinates. P = ((2 × 2 + 1 × 8) ÷ 3, (2 × 1 + 1 × 7) ÷ 3) = (4, 3).
By vectors. AB = (6, 6), so P = A + ⅓AB = (2, 1) + (2, 2) = (4, 3).
Part (b) asks for an equation. The gradient of AB is 6 ÷ 6 = 1, so the perpendicular gradient is −1. The line is y − 3 = −(x − 4), so x + y = 7. Check: 4 + 3 = 7.
A model reason: “I used coordinates because part (b) asks for the equation of a line, which needs a gradient and a point.”
The mistake that costs marks
A common slip is to start in vectors and then drop back into coordinates without saying so. The working mixes both notations and the marker cannot follow it.
| Approach | Line of working | Problem |
|---|---|---|
| Mixed | AB = (6, 6), so gradient of AB = 6 | The vector (6, 6) is read as if it were one number |
| Consistent | AB = (6, 6), so gradient = 6 ÷ 6 = 1 | Vector used only to read the step |
A vector (6, 6) does not have a gradient of 6. Choose one method at the start and keep its notation, or state clearly when you change.
Check yourself
For each question, choose coordinates or vectors and give a reason. (a) Find the area of a triangle with given vertices. (b) Given OA = a and OB = b, show that the midpoint M of AB has position vector ½(a + b). (c) Find the equation of the line through M perpendicular to AB.
Answer
(a) Coordinates, because the area comes directly from the vertices.
(b) Vectors, because the data is in letters and there are no coordinates to use.
(c) Coordinates, because the question asks for an equation, which needs a gradient and a point.
What to study next
Go on to checking an algebraic intersection against a geometric restriction, then the geometry problems practice set.
The word-problem structure worksheet helps you find the signal words. A teacher can ask you for your reasons in online one-to-one Additional Mathematics tuition.