22.3 Vectors in geometric problems and proofs — Learn

Prerequisites: vector notation, addition, subtraction and scalar multiplication.

Objectives: express routes between points using vectors; construct a geometric argument and prove a midpoint result.

Choose a route between points

In a diagram, a route from one point to another can pass through other points. Add the vectors along the route. Reversing a route changes the sign of its vector. A position vector is the vector from a chosen origin to a point.

Example 1. In triangle $ABC$ , let $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{BC}=\mathbf{v}$ . Find $\overrightarrow{AC}$ and $\overrightarrow{CA}$ .

\begin{aligned} \overrightarrow{AC} &= \overrightarrow{AB}+\overrightarrow{BC} \\ &=\mathbf{u}+\mathbf{v} \\ \overrightarrow{CA} &=-\overrightarrow{AC} \\ &=-\mathbf{u}-\mathbf{v} \end{aligned}

Build a proof from the definition

A midpoint divides a line segment into two equal parts. Let $ABCD$ be a parallelogram with $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{AD}=\mathbf{v}$ . Opposite sides have equal directed vectors, so $\overrightarrow{BC}=\mathbf{v}$ and $\overrightarrow{DC}=\mathbf{u}$ .

Example 2. Show that the diagonals of a parallelogram bisect each other.

Choose $A$ as the origin. Then the position vectors of $B$ , $D$ and $C$ are $\mathbf{u}$ , $\mathbf{v}$ and $\mathbf{u}+\mathbf{v}$ .

\begin{aligned} \text{midpoint of }AC &= \frac{\mathbf{0}+\mathbf{u}+\mathbf{v}}{2} \\ &=\frac{1}{2}\mathbf{u}+\frac{1}{2}\mathbf{v} \\ \text{midpoint of }BD &= \frac{\mathbf{u}+\mathbf{v}}{2} \\ &=\frac{1}{2}\mathbf{u}+\frac{1}{2}\mathbf{v} \end{aligned}

The diagonals have the same midpoint, so they bisect each other. Each midpoint step follows from adding the endpoint position vectors and halving.

Watch out. Equal lengths alone do not prove that two segments have the same midpoint. Compare both midpoint position vectors, including both components.

Progress check

For the chapter problem, the midpoint of $AC$ has position vector $\frac12(\mathbf{u}+\mathbf{v})$ . Work out the midpoint of $BD$ as a position vector and compare the two. Does this prove that the diagonals bisect each other for every parallelogram? Explain.

You should now be able to

find a vector by adding the vectors along a route;
reverse a route by changing the vector's sign;
express points in terms of position vectors from a chosen origin;
construct a vector proof by comparing expressions for the same geometric point.