## 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 2vec v. In general, when we have two vectors vec u and vec v such that

vec u = kvec v,

we say that vec u and vec 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, they will have different magnitudes. If k = 2, then vec u will be twice as long as vec v.

Here are a few examples of scalar multiplication:

Scalar multiplication is distributive, meaning

k(vec u + vec v) = kvec u + kvec v qquad and qquad (a + b)vec u = avec u + bvec u,

and it is also associative:

a(bvec u) = (ab)vec u.

Like with addition, there is an identity for this operation as well: the multiplicative identity. This is just the scalar 1, because 1vec 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:

hat u = (vec u)/(|vec u|).

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 vec u gives us hat u (the hat denotes a unit vector).