Question

Find a unit vector, u, with the same direction?

As v=(-2,4)

Answer 1

`(-1/sqrt5, 2/sqrt5)`

Explanation:

Any vector in the same direction of a vector `vec v` has to be of the form `a vec v` where `a>0` is a scalar. If this resulting vector is to be a unit vector, we must have

`||a vec v|| = |a|*||vec v||=1 implies a = 1/||vec v||`

So, a unit vector in the same direction as a given vector `vec v` is given by

`hat v = (vec v)/||vec v||`

In our case `vec v = (-2,4)`

So `||vec v|| = sqrt((-2)^2+4^2)=sqrt 20 = 2sqrt5`

Thus, the unit vector we want is

`1/(2sqrt5)(-2,4) = (-1/sqrt5, 2/sqrt5)`