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 $\displaystyle 2 \vec{v}$. In general, when we have two vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ such that

$\displaystyle \vec{u} = k \vec{v}$,

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

Here are a few examples of scalar multiplication:

Scalar multiplication is distributive, meaning

$\displaystyle k \left ( \vec{u} + \vec{v} \right ) = k \vec{u} + k \vec{v} \qquad \allowbreak\quad\allowbreak\text{and}\allowbreak\quad\allowbreak \qquad \left ( a + b \right ) \vec{u} = a \vec{u} + b \vec{u}$,

and it is also associative:

$\displaystyle a \left ( b \vec{u} \right ) = \left ( a b \right ) \vec{u}$.

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

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:

$\displaystyle \hat{u} = \frac{\vec{u}}{\left \lvert \vec{u} \right \rvert}$.

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 $\displaystyle \vec{u}$ gives us $\displaystyle \hat{u}$ (the hat denotes a unit vector).