Vectors (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
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Vectors (AS Unit 1: Pure Mathematics A) explained
The position vector of a point A relative to a fixed origin O is denoted by r_A or OA, representing the displacement from the origin to A.
Read the full explanation
If point A has Cartesian coordinates (x₁, y₁) and point B has coordinates (x₂, y₂), their position vectors are OA = x₁ i + y₁ j and OB = x₂ i + y₂ j. The relative displacement vector from point A to point B is calculated by vector subtraction: AB = OB − OA = (x₂ − x₁)i + (y₂ − y₁)j. The straight-line Euclidean distance between points A and B equals the magnitude of the displacement vector AB, calculated using Pythagoras' theorem: |AB| = √((x₂ − x₁)² + (y₂ − y₁)²). The midpoint M of the line segment AB has position vector OM = ½(OA + OB) = ½(x₁ + x₂)i + ½(y₁ + y₂)j.
Your focus
- Distinguish between position vectors relative to an origin and displacement vectors between points.
- Calculate displacement vectors using the relation AB = OB − OA.
- Calculate distances between points by finding the magnitude of their displacement vector.
Vectors (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- finding the displacement vector AB using OB − OA
- calculating the distance between two points as the magnitude of AB
- calculating the position vector of a midpoint or point dividing a segment in a given ratio
Examiner Tips
- 💡Remember the nose-to-tail displacement rule: AB = OB − OA.
- 💡The distance between points A and B is simply the magnitude of the vector AB: |AB|.
Common Mistakes
- calculating AB as OA − OB instead of OB − OA, reversing the displacement direction
- confusing position vectors (anchored at the origin) with free displacement vectors