Chapter 22: Vectors

Vectors describe both a direction and a distance. You can use them to describe translations, calculate journeys, and explain why geometric facts are true.

The chapter problem

Diagonals of a parallelogram

Start at point $A$. Move by vector $\mathbf{u}$ to point $B$, then by vector $\mathbf{v}$ to point $C$. A second route from $A$ moves by $\mathbf{v}$ to $D$. The four points form a parallelogram when opposite sides have the same direction and length.

First contact — no working yet. Predict: do the diagonals $AC$ and $BD$ meet halfway along each diagonal? Say what you think, or sketch a parallelogram.

By the end, you will use vectors to give a proof and check your prediction.

By the end of this chapter

You will be able to:

• describe a translation using a two-dimensional vector;
• write and draw vectors as columns, and interpret their horizontal and vertical changes;
• add, subtract and multiply vectors by a scalar;
• use vector notation to construct a geometric argument and proof.

Roadmap

1. 22.1 Vector notation and column vectors — describe translations and connect arrows to columns.
2. 22.2 Adding and subtracting vectors — combine journeys and scale a vector.
3. 22.3 Vectors in geometric problems and proofs — express points in terms of vectors and prove geometric facts.
4. Review — return to the parallelogram and mix the methods.

Your turn to connect it

Maps, movement and design can involve directions and distances. Where might a description of movement help you or someone you know? Write one example in your own words. You can return to it after the chapter.