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

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

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

Here are a few examples of scalar multiplication:

A few scalar multiples of vector u\displaystyle \vec{u}: 1, 2, −1, 0.5, and −1.5

Scalar multiplication is distributive, meaning

k(u+v)=ku+kvand(a+b)u=au+bu\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:

a(bu)=(ab)u\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 1u=u\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:

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