Year 10 Higher · Vectors
Understand what a vector describes
Learn how to describe journeys through shapes with vectors and spot when two vectors are parallel.
Definition
A vector describes a movement with size and direction, written as a column: across on top, up underneath. On the Higher paper vectors also describe journeys around shapes: going from A to B and then B to C is the same as going straight from A to C, so routes can be added like arithmetic.
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. Add vectors by following one movement after another: the route A to B to C equals the direct route A to C.
- 2. Going backwards along a vector flips its sign: the journey B to A is minus the journey A to B.
- 3. Multiply a vector by a number to scale it; multiply by a negative number to scale and reverse it.
- 4. Two vectors are parallel exactly when one is a multiple of the other: (4, 6) is parallel to (2, 3).
- 5. In a four-sided shape, opposite sides that are equal vectors prove the sides parallel and equal in length.
Remember:
- •Follow the journey letter by letter; backwards means minus; parallel means one is a multiple of the other.
Don’t Forget...
- •Writing the journey B to A as p instead of -p. Direction matters: reversing a journey reverses the vector.
- •Claiming vectors are parallel without showing one is a multiple of the other. Parallel needs proportion in both parts: (4, 6) and (2, 3) are parallel because both parts double; (4, 6) and (2, 4) are not.
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 quadrilateral ABCD, the journey A to B is the vector p and the journey A to D is q. M is the midpoint of BD. Write the journey B to D, and then A to M, in terms of p and q.
B to D goes backwards along p to A, then along q to D: so B to D = -p + q, usually written q - p.
M is halfway along BD, so B to M is half of (q - p).
A to M follows A to B, then B to M: p + half of (q - p).
Simplify: p + (12)q - (12)p = (12)p + (12)q.
Answer: B to D = q - p, and A to M = (12)(p + q)
Animated Example:
Free Understand what a vector describes Worksheet
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Next: Vectors (Year 11 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