Skill: Multiply vectors by scalars

To multiply a vector by a scalar, multiply every component of the vector by the same real number.

If

$$\mathbf{v}=\begin{pmatrix}a\\b\end{pmatrix}$$

and the scalar is $k$, then

$$k\mathbf{v} = k\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ka\\kb\end{pmatrix}.$$

Worked example

Find $-2\mathbf{v}$ if

$$\mathbf{v}=\begin{pmatrix}3\\-4\end{pmatrix}.$$

Step 1: Multiply the first component by $-2$.

$$-2(3)=-6$$

Step 2: Multiply the second component by $-2$.

$$-2(-4)=8$$

Step 3: Write the new vector.

$$-2\mathbf{v} = -2\begin{pmatrix}3\\-4\end{pmatrix} = \begin{pmatrix}-6\\8\end{pmatrix}.$$

So,

$$\boxed{-2\mathbf{v}=\begin{pmatrix}-6\\8\end{pmatrix}}.$$

The scalar $-2$ has two effects:

• The magnitude is multiplied by $\vert -2 \vert =2$, so the vector is twice as long.
• Because the scalar is negative, the direction is reversed.

A positive scalar keeps the direction the same, while multiplying by $0$ gives the zero vector:

$$0\mathbf{v}=\begin{pmatrix}0\\0\end{pmatrix}.$$