Question

How do you find a unit vector u in the same direction as the vector ⟨1,−2,−3⟩?

Answer 1

`vec v = { {1,-2,-3}}/sqrt(14)`

Explanation:

A unit vector is a vector `vec u` such that

`norm vec u = 1`

Given a vector `vec V ={1,-2,-3}` the way to find its associated unit vector `vec v` is normalizing it. Then

`vec v = vec V/(norm vec V) ={ {1,-2,-3}}/sqrt(1^2+(-2)^2+(-3)^2} = { {1,-2,-3}}/sqrt(14)`.

We know that `<< vecV,vec V >> = norm(vec V)^2` then

`<< vec v, vec v >> = << vec V/(norm vec V) , vec V/(norm vec V) >> = << vec V, vec V>>/norm(vec V)^2 = norm(vecV)^2/norm vecV^2 = 1`