Vector proof questions ask you to show that points are collinear or that lines are parallel, using vector algebra. They look intimidating but follow a clear pattern once you've practised them.
Parallel vectors: $\overrightarrow{AB}$ is parallel to $\overrightarrow{CD}$ if one is a scalar multiple of the other — i.e. $\overrightarrow{AB} = k\overrightarrow{CD}$ for some value $k$.
Collinear points: Three points $A$, $B$, $C$ are collinear (on the same straight line) if $\overrightarrow{AB}$ is a scalar multiple of $\overrightarrow{AC}$ and they share a common point. You need both conditions.
General approach:
- Express each vector path in terms of the given vectors a and b
- Use $\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB}$ (go via the origin if needed)
- Simplify and factorise
- State your conclusion explicitly — "therefore $\overrightarrow{AB}$ is parallel to $\overrightarrow{CD}$"
Example: If $M$ is the midpoint of $AB$, then $\overrightarrow{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$.
Don't just get the vectors — write the conclusion in words. Marks are often lost by students who do the algebra correctly but forget to state what it means.