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 . 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).