1. ## Orthogonal Projection

Find the orthogonal projection of (x,y,z) onto the subspace of R^3 spanned by vectors (1,2,2),(-2,2,-1).

what i did was
[(x,y,z).(1,2,2)](1,2,2) + [(x,y,z).(-2,2,-1)](-2,2,-1)
which is equals to
(x+2y+2z,2x+4y+4z,2x+4y+4z) + (4x-4y+2z,-4x+4y-2z,2x-2y+z)
and then that is equals to
(5x-2y+4z,-2x+8y+2z,4x+2y+5z)

but the answer given is 1/9(5x-2y+4z,-2x+8y+2z,4x+2y+5z)
which is the same as what i got except the 1/9 in front. can anyone tell me what i did wrong?

another question,
if i am given 2 vectors X & Y in R^3 and given the inner product of <X,Y>
how do i calculate the angle between X and Y? i know how to calculate ||X|| and ||Y|| which is just <X,X> and <Y,Y> right?

2. Originally Posted by yen yen
Find the orthogonal projection of (x,y,z) onto the subspace of R^3 spanned by vectors (1,2,2),(-2,2,-1).

what i did was
[(x,y,z).(1,2,2)](1,2,2) + [(x,y,z).(-2,2,-1)](-2,2,-1)
which is equals to
(x+2y+2z,2x+4y+4z,2x+4y+4z) + (4x-4y+2z,-4x+4y-2z,2x-2y+z)
and then that is equals to
(5x-2y+4z,-2x+8y+2z,4x+2y+5z)

but the answer given is 1/9(5x-2y+4z,-2x+8y+2z,4x+2y+5z)
which is the same as what i got except the 1/9 in front. can anyone tell me what i did wrong?

another question,
if i am given 2 vectors X & Y in R^3 and given the inner product of <X,Y>
how do i calculate the angle between X and Y? i know how to calculate ||X|| and ||Y|| which is just <X,X> and <Y,Y> right?

What you did is ALMOST correct, but you must work with an orthonormal basis of the subspace! In general this would imply to do Gram-Schmidt on a certain basis of the subspace, but in this particular case it is much easier since the basis elements are orthogonal so we only have to divide each by its length (= its norm), and then the basis we work with should be $\displaystyle \frac{(1,2,2)}{\|(1,2,2)\|}=\frac{1}{3}(1,2,2)\,,\ ,\,\frac{(-2,2,-1)}{\|(-2,2,-1)\|}=\frac{1}{3}(-2,2,-1)$ ...and now do what you did BUT with the two normalized elements of the basis instead (can you see now from where that $\displaystyle 1\slash 9$ came?)

Tonio