O-Level E-Math and SEC G3 Mathematics K310
G7: Vectors in two dimensions
Express routes with consistent direction, simplify vector relations, and justify collinearity or parallelism from the result.
Core notes
A vector records magnitude and direction. Express every route with consistent arrow direction, use position vectors relative to one origin, and compare coefficients to establish geometric relationships.
Notation and directed segments
A column vector gives horizontal and vertical components. A directed segment such as points from to , so reversing the endpoints changes the sign. A translation moves every point by the same vector.
Position vectors and magnitude
The position vector of a point runs from the chosen origin to that point. Subtract position vectors in endpoint-minus-startpoint order. The magnitude of is .
Vector algebra and geometry
Add and subtract corresponding components or express routes in terms of two coplanar basis vectors. Scalar multiplication changes magnitude and may reverse direction. Parallel vectors are scalar multiples; collinearity also requires the relevant points to lie on one line.
Formulae and methods
| Relationship | Use |
|---|---|
| Find a directed segment from position vectors. | |
| Find vector magnitude. |
Worked examples
Example 1: Point has position vector and has . Find and its magnitude.
- Subtract start from endpoint.
- .
- Magnitude is .
Answer: and its magnitude is .
Example 2: Given , find .
- Reverse the vector direction.
- Multiply the full expression by .
- Distribute the negative sign.
Answer: .
Chapter checkpoint
- Distinguish a point from the displacement vector that reaches it.
- Add vectors head to tail and reverse direction before subtraction.
- Use one common vector basis when testing collinearity, parallelism or a route identity.
Exam traps and retrieval check
Avoid these traps
- Subtracting position vectors in the wrong order.
- Adding components when the question asks for magnitude.
- Claiming vectors are parallel without one consistent scalar multiplying every component.
Check from memory
How is related to ?
.
What does a negative scalar do to a vector?
It reverses direction and scales the magnitude by the scalar's absolute value.
What is a position vector measured from?
The chosen origin.
Official K310 coverage
This chapter maps to 8 official outcomes: G7.1, G7.2, G7.3, G7.4, G7.5, G7.6, G7.7, G7.8. Check the official 2027 K310 syllabus for exact assessable wording and paper details.