Column Vectors

Written as [x, y]. Top number is horizontal movement, bottom is vertical.

Example: [3, -2] means 3 right, 2 down.

Adding & Subtracting

Just add/subtract the top numbers and the bottom numbers separately.

Example: [1, 2] + [3, 4] = [4, 6]

Scalar Multipliers

Multiply both numbers by the scalar.

Example: 3 × [2, 5] = [6, 15]

Resultant Vectors

The single vector that replaces multiple movements. Geometrically, it's the "shortcut".

Geometric Vectors

Vectors like a and b on a diagram. Go "the long way round" using known paths.

Midpoints & Ratios

Find points a fraction of the way along a line. E.g. M is the midpoint of AB: OM = OA + 0.5AB.

Parallel Vectors

Vectors are parallel if one is a multiple of the other.

Example: [2, 3] is parallel to [4, 6] because [4, 6] = 2 × [2, 3].

Collinear Points

To prove points A, B, and C lie on a straight line: show AB is parallel to BC and they share point B.