Overview
Vectors have both magnitude and direction — used for displacement, velocity and geometry.
Basics
- Written as a column ⎛x⎞⎝y⎠ or as x i + y j.
- Magnitude (length) = √(x² + y²).
- A unit vector has magnitude 1: divide a vector by its magnitude.
- A position vector locates a point relative to the origin.
Operations
- Add/subtract by combining components.
- A scalar multiple ka keeps the direction (or reverses it) and scales the length.
- Two vectors are parallel if one is a scalar multiple of the other.
Worked example
If a = (3, 4), |a| = √(9 + 16) = 5, so the unit vector is (0.6, 0.8).
Velocity & position
position = initial position + velocity × time, e.g. r = r₀ + t v.
Common mistakes
- Adding magnitudes instead of components when adding vectors.
Exam tips
- "Parallel" → check if the components are in the same ratio.