Vector Calc: Ortho Projections

[winowmak3r]

Homework Statement

Find the projection of P in the direction of Q and the component of P orthogonal to Q.

P=i-3j+k Q=-i+2j+5k

Homework Equations

Proj_Q(P)={(P*Q)/(Q*Q)}Q

Orth_Q(P)=P-Proj_Q(P)

The Attempt at a Solution

First I get P=(1,-3,1) and Q=(-1,2,5)

Proj_Q(P)
=(-2/30)*(-1,2,5)
={(1/15),(-2/15),(-1/3)}

That's correct, as far as the book is concerned.

Now, finding the orthogonal part is where I get held up.

I do this:
Orth_Q(P)=
=(1,-3,1)-{(1/15),(-2/15),(-1/3)}
={14/15,(47/15),(4/3)}

Which is wrong. The book says the correct answer is: {(74/15),(32/15),(-2/3)}.

I'm at a complete loss as to how they got this. Any help is much appreciated.

 

[gabbagabbahey]

Your formula for $\text{Proj}_\textbf{Q}(\textbf{P})$ should have $\textbf{Q}\cdot\textbf{Q}$ in the denominator, not $\textbf{P}\cdot\textbf{P}=11$, although you seem to have done the calculations correctly.

I think the answer given in the back of your text for the second part is wrong, and they switched to using $\textbf{P}=5\textbf{i}+2\textbf{j}-\textbf{k}$ by mistake in that 2nd part to get their (incorrect) answer.

 

[jbunniii]

Proj_Q(P)={(P*Q)/(P*P)}Q

I assume you mean

Proj_Q(P)={(P*Q)/(Q*Q)}Q

which must be what you used since you got the right answer for Proj_Q(P).

For Orth_Q(P), I get (14, -43, 20)/15, which doesn't match either your answer or the book's.

 

[winowmak3r]

Ah, yes. It should be QQ in the denominator rather than PP. I just translated it incorrectly to the forum, I've been working on this for a while now and am getting frustrated.

But if they're using P=5i+2j-k...that's not in the problem. It appears the textbook is incorrect?

 

[gabbagabbahey]

Yes, the textbook is incorrect. However,as jbunnii pointed out, your answer is also a little off (pay close attention to your negative signs)

 

[winowmak3r]

I assume you mean

Proj_Q(P)={(P*Q)/(Q*Q)}Q

which must be what you used since you got the right answer for Proj_Q(P).

For Orth_Q(P), I get (14, -43, 20)/15, which doesn't match either your answer or the book's.

Alright, I realize now what I did was incorrect. OrthoP should be = (14, -43, 20)/15, I forgot that there was a -1/3 in ProjP which is why I didn't get 20/15 and added the -2 wrong to 45.

However, that's still not what's in the back of the book. Argh. I guess I can take my case to the prof in the morning. Thanks guys.