To make it easier to refer to them, let's call the first vector vec u and the second vec v. We want the project of vec u onto vec v:
proj_vec v vec u = ((vec u*vec v)/||vec v||^2)*vec v
That is, in words, the projection of vector vec u onto vector vec v is the dot product of the two vectors, divided by the square of the length of vec v times vector vec v. Note that the piece inside the parentheses is a scalar that tells us how far along the direction of vec v the projection reaches.
First, let's find the length of vec v:
||vec v||=sqrt(3^2+2^2+(-3)^2) = sqrt22
But note that in the expression what we actually want is ||vec v||^2, so if we square both sides we just get 22.
Now we need the dot product of vec u and vec v:
vec u * vec v = (1xx3+(-2)xx2+3xx(-3)) = (3-4-9)= (-10)
(to find the dot product we multiply the coefficients of i, j and k and add them)
Now we have everything we need:
proj_vec v vec u = ((vec u*vec v)/||vec v||^2)*vec v = (-10/22)(3i+2j−3k)
=(-30/22i-20/22j+30/22k) = (-15/11i-10/11j+15/11k)
