Scalar Multiplication

Another very basic operation on vectors is scalar multiplication. For a basic introduction, you can read through my short lesson about scalars.

Vectors can be multiplied by a scalar, such as $2$ or $-1$ or any number $c$ . Scalar multiplication by a scalar $c$ is defined as

c\vec{v} = \begin{bmatrix} cv_1 \\ cv_2 \end{bmatrix}.

Let us illustrate this through an example:

\vec{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}.

We now multiply this vector by $2$ :

2\vec{v} = \begin{bmatrix} 2 \cdot 3 \\ 2 \cdot 4 \end{bmatrix} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}.

Graphical Representation

I have touched upon the idea of “scaling” a vector in my introductory lesson about scalars. This is because, when depicted graphically, the vector $c\vec{v}$ , is simply a scaled version of $\vec{v}$ :

Multiplying a vector by a negative number (say $-1$ ), simply reverses its direction: