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:

uu2uu12u32
A few scalar multiples of vector `vec u`: 1, 2, −1, 0.5, and −1.5

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).