Year 11 Higher · Vectors
Solve a problem using vectors
Learn how to use vectors and ratios to solve geometric problems, show that three points lie on a straight line and build complete vector proofs.
Definition
A vector describes a movement with size and direction, and on the Higher paper it becomes a tool for proving geometric facts. Any journey through a labelled diagram can be written in terms of two given vectors, usually a and b, and the expressions that result obey ordinary algebra. Because multiplying a vector by a number always keeps it parallel to the original (a negative number simply reverses it along the same line), showing that one vector is a multiple of another proves the two are parallel, and that single idea drives every proof in this unit.
Why Do I Have to Learn This?
Vectors describe movement with both size and direction. They steer ships, planes and characters in video games.
Key Rules:
- 1. Write any journey in terms of the given vectors by following a route through labelled points: A to C equals A to B followed by B to C, and travelling backwards along a vector changes its sign.
- 2. Simplify vector expressions with ordinary algebra, collecting the a terms and the b terms separately.
Remember:
- •Write the journey, turn the ratio into a fraction, then factorise: a scalar multiple proves parallel, and parallel through a shared point proves a straight line.
Don’t Forget...
- •Reading AP:PB = 1:2 as A to P being 12 of A to B. The ratio splits AB into 1 + 2 = 3 equal parts, so A to P is 13 of A to B; the fraction always comes from the whole line, never from the other share.
- •Ending a straight-line proof at the algebra, for example stopping at A to C = 3(A to B). The multiple only shows the two vectors are parallel; the proof must also say that both pass through A before concluding the points are collinear.
- •Writing the journey A to B as OA - OB. The route goes backwards along OA and then out along OB, so A to B = OB - OA; the wrong order gives the reverse vector B to A and every sign in the proof comes out wrong.
Translation
Sliding a shape to a new position without turning it.
Moving a shape 3 squares to the right is a translation.
Worked Example:
In triangle OAB, the journey O to A is the vector a and the journey O to B is b. The point P lies on AB with AP:PB = 1:2. Find the journey O to P in terms of a and b.
Write A to B as a route through O: backwards along a, then out along b, so A to B = b - a.
The ratio 1:2 cuts AB into 3 equal parts, so A to P = (13)(b - a).
Follow the route O to A to P: O to P = a + (13)(b - a).
Collect terms: a - (13)a + (13)b = (23)a + (13)b.
Answer: O to P = (23)a + (13)b
Why It Works:
Vector algebra is valid because each component obeys ordinary arithmetic, so expressions in a and b can be expanded, factorised and collected exactly like algebra. Multiplying a vector by a scalar changes its length but keeps it along the same line, reversing it if the scalar is negative, which is precisely what parallel means, and two parallel vectors that share a point must lie along the same line, which is precisely what collinear means.
Another Example:
Relative to an origin O, the points A, B and C satisfy O to A = a + b, O to B = 3a + 2b and O to C = 7a + 4b. Show that A, B and C lie on a straight line.
Find A to B by going back to O and out to B: A to B = (3a + 2b) - (a + b) = 2a + b.
Find A to C the same way: A to C = (7a + 4b) - (a + b) = 6a + 3b.
Factorise: 6a + 3b = 3(2a + b), so A to C = 3 x (A to B).
A to C is a scalar multiple of A to B, so they are parallel, and both journeys start at A, so A, B and C lie on one straight line.
Answer: A to C = 3(A to B), so the three points are collinear
Animated Example:
Free Solve a problem using vectors Worksheet
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Before this: Vectors (Year 10 Higher)
Let’s Practise the Concept
Step 1 of 3
Put them in order
Tap the numbers from smallest to largest.
In order so far: nothing yet