Skill: Add and subtract vectors

Vectors can be added or subtracted by working with their corresponding components.

• Add horizontal components together and vertical components together.
• For subtraction, subtract each corresponding component. This is the same as adding the opposite vector:

$$\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})$$

Worked example

Let

$$\mathbf{a}=\begin{pmatrix}3\\4\end{pmatrix}, \qquad \mathbf{b}=\begin{pmatrix}-1\\2\end{pmatrix}.$$

Find $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$.

1. Add the vectors

Add the corresponding components:

$$\mathbf{a}+\mathbf{b} = \begin{pmatrix}3\\4\end{pmatrix} + \begin{pmatrix}-1\\2\end{pmatrix} = \begin{pmatrix}3+(-1)\\4+2\end{pmatrix} = \begin{pmatrix}2\\6\end{pmatrix}.$$

So, the resultant vector is

$$\boxed{\mathbf{a}+\mathbf{b}=\begin{pmatrix}2\\6\end{pmatrix}}.$$

Geometrically, this means place $\mathbf{b}$ at the head of $\mathbf{a}$. The resultant goes from the tail of $\mathbf{a}$ to the head of the moved vector $\mathbf{b}$.

2. Subtract the vectors

Subtract the corresponding components:

$$\mathbf{a}-\mathbf{b} = \begin{pmatrix}3\\4\end{pmatrix} - \begin{pmatrix}-1\\2\end{pmatrix} = \begin{pmatrix}3-(-1)\\4-2\end{pmatrix} = \begin{pmatrix}4\\2\end{pmatrix}.$$

Therefore,

$$\boxed{\mathbf{a}-\mathbf{b}=\begin{pmatrix}4\\2\end{pmatrix}}.$$

Equivalently, reverse the direction of $\mathbf{b}$:

$$-\mathbf{b}=\begin{pmatrix}1\\-2\end{pmatrix},$$

then add:

$$\mathbf{a}-\mathbf{b} = \mathbf{a}+(-\mathbf{b}) = \begin{pmatrix}3\\4\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix} = \begin{pmatrix}4\\2\end{pmatrix}.$$

The vector $\begin{pmatrix}4\\2\end{pmatrix}$ describes both a direction and a magnitude. Its magnitude is

$$\left \vert \mathbf{a}-\mathbf{b}\right \vert = \sqrt{4^2+2^2} = \sqrt{20} = 2\sqrt{5}.$$

Remember that addition is commutative, so $\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}$. Subtraction is not commutative: usually $\mathbf{a}-\mathbf{b}\ne\mathbf{b}-\mathbf{a}$.