22.2 Adding and subtracting vectors — Learn

Prerequisites: column vectors and directed movement from section 22.1.

Objectives: add and subtract vectors; multiply a vector by a scalar; show these operations in diagrams and columns.

Add vectors component by component

When two movements happen in sequence, add their horizontal changes and add their vertical changes. This is vector addition.

Example 1. Work out $\begin{pmatrix}3\\-2\end{pmatrix}+\begin{pmatrix}-5\\4\end{pmatrix}$.

The combined movement is 2 left and 2 up.

Subtract vectors and multiply by a scalar

Subtracting a vector means adding its opposite. Multiplying by a scalar—a number that scales a vector—multiplies each component.

Example 2. Work out $2\begin{pmatrix}1\\3\end{pmatrix}-\begin{pmatrix}4\\-2\end{pmatrix}$.

Watch out. The lower component is $6-(-2)=8$, not $4$. Subtracting a negative number increases the result. Check by adding the opposite vector.

Progress check

For the chapter problem, the route from $A$ to $C$ is $\mathbf{u}+\mathbf{v}$, and the other route is $\mathbf{v}+\mathbf{u}$. Calculate both with $\mathbf{u}=\begin{pmatrix}3\\-2\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}-1\\5\end{pmatrix}$. What do you notice, and what does it tell you about the parallelogram?

You should now be able to

• add and subtract vectors by operating on matching components;
• multiply each component by a scalar;
• interpret a negative component as movement in the opposite direction;
• represent a vector calculation with a diagram or a column.