Scalar multiplication means multiplying a vector by a scalar. To do this, we simply put a scalar coefficient in front of the vector, like . In general, when we have two vectors and such that
we say that and are scalar multiples of each other. They will always be parallel to each other—if is positive, they will have the same direction; if is negative, they will have opposite directions. Unless , they will have different magnitudes. If , then will be twice as long as .
Here are a few examples of scalar multiplication:
Scalar multiplication is distributive, meaning
and it is also associative:
Like with addition, there is an identity for this operation as well: the multiplicative identity. This is just the scalar 1, because .
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:
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 gives us (the hat denotes a unit vector).