22.1 Vector notation and column vectors — Learn

Prerequisites: coordinates, directed movement, and the horizontal and vertical axes.

Objectives: describe translations as two-dimensional vectors; read and draw column vectors on a grid.

Describe a translation

A translation moves every point of a shape the same distance in the same direction. A vector records that movement. Its direction matters as well as its length.

We write a column vector with the horizontal change above the vertical change. Right and up are positive; left and down are negative.

This vector means move 3 units right and 2 units down.

Example 1. Point $P$ moves from $(1,2)$ to $(5,-1)$. Describe the translation.

The movement is 4 units right and 3 units down.

Predict: before moving the controls, predict where the point will move when the horizontal entry is negative. Write your prediction down first.

Test: test your prediction with the figure.

Explain: how do the signs show the direction? Write your explanation before reading on.

Column vectors

The same vector can be shown by an arrow on a square grid or by a column. Count the horizontal and vertical steps from the arrow's start to its end.

Example 2. An arrow starts at $(2,-1)$ and ends at $(-1,3)$. Write its column vector.

So the arrow moves 3 units left and 4 units up. The grid representation and the column representation describe the same movement.

Watch out. Reversing the column gives the opposite movement. Check the start and end points: reversing an arrow changes both signs.

Progress check

In the chapter problem, describe the route from $A$ to $C$ using the two side vectors. We will use this notation to compare the diagonals in section 22.3.

You should now be able to

• describe a translation as a two-dimensional vector;
• read a vector from two endpoints by subtracting coordinates;
• draw a column vector as a directed arrow on a square grid;
• explain what each component says about horizontal and vertical movement.