Scalar multiplication

Scalar multiplication means multiplying a vector by a scalar. To do this, we simply put a scalar coefficient in front of the vector, like $2\overrightarrow{v}\text{.}$ In general, when we have two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ such that

$\overrightarrow{u}=k\overrightarrow{v}\text{,}$

we say that $\overrightarrow{u}$ and $\overrightarrow{v}$ are scalar multiples of each other. They will always be parallel to each other—if $k$ is positive, they will have the same direction; if $k$ is negative, they will have opposite directions. Unless $k=1\text{,}$ they will have different magnitudes. If $k=2\text{,}$ then $\overrightarrow{u}$ will be twice as long as $\overrightarrow{v}\text{.}$

Here are a few examples of scalar multiplication:

Scalar multiplication is distributive, meaning

$k(\overrightarrow{u}+\overrightarrow{v})=k\overrightarrow{u}+k\overrightarrow{v}$ and $(a+b)\overrightarrow{u}=a\overrightarrow{u}+b\overrightarrow{u}\text{,}$

and it is also associative:

$a(b\overrightarrow{u})=(ab)\overrightarrow{u}\text{.}$

Like with addition, there is an identity for this operation as well: the multiplicative identity. This is just the scalar 1, because $1\overrightarrow{u}=\overrightarrow{u}\text{.}$

A unit vector is a vector whose magnitude is one. We can use scalar multiplication to normalize any vector (turn it into a unit vector) like so:

$\hat{u}=\frac{\overrightarrow{u}}{\left|\overrightarrow{u}\right|}\text{.}$

All we have to do is divide by the vector’s magnitude (multiply by the reciprocal). This will give us a vector of length one that still points in the original direction. Normalizing $\overrightarrow{u}$ gives us $\hat{u}$ (the hat denotes a unit vector).