Question

Find the projection of u onto v, then write u as the sum of two orthogonal vectors.
u=12,16 v= -6,-4
Possible answers:
<6,8>
<-6,-4>
<12,16>
none of these


I have done the problem and my answer is nowhere near the others so I assume I must be wrong.

Answer 1

What did you get as an answer?

Answer 2

what is the equation for the projection of one vector onto another?

Answer 3

(v*u/||v||^2)v

Answer 4

i completed that with 15.692307692307692307692307692308, 10.461538461538461538461538461538

Answer 5

read this, my wife is yelling at me to stop

Answer 6

https://answers.yahoo.com/question/index?qid=20090712193147AA0T9WX

Answer 7

The part i think im stuggling with is the "sun of the orthagaonal" part

Answer 8

So you have the formula correct. So we can go piece by piece then. So what would u dot v be?

Answer 9

-136

Answer 10

Alright, cool. And then ||v||^2 ?

Answer 11

52

Answer 12

Yep. So we would have (-136/52)v. We can reduce that to (-34/13)v. So I'm guessing you got something along the lines of that and it weirded ya out?

Answer 13

no... we are still on the path i originally took
-34/13*(-6,-4)= 15.692307692307692307692307692308, 10.461538461538461538461538461538

Answer 14

"then write u as the sum of two orthogonal vectors" is my issue
i also forgot to include the line " one of the vectors being the projection

Answer 15

Yeah, and you did nothing wrong. I wouldnt go and turn everything into decimals, though, usually best to keep it as fractions. So your projection would be (204/13, 136/13) When it comes to having two orthogonal vectors, we can say u is the sum of two orthogonal vectors, say v1 and v2. So u = v1 + v2
v1 is simply the projection we just found. So that leaves v2. Well, algebraically, we can just subtract v1 from both sides and say v2 = u - v1. So the two vectors we want are:
1. The projection we just found
2. u minus the projection we find.

So given we have the projection of (204/13, 136/13), we want to find (12,16) - (204/13, 136/13)

Answer 16

(-64/13, 74/13)?

Answer 17

Well, we would subtract corresponding components. So we would have 12 - 204/13. This is 156/13 - 204/13 = (156-204)/13 = -48/13
The second component would be done the same way, youd have 16 - 136/13

Answer 18

.... i mistyped both of those i meant (-48/13, 72/13)

Answer 19

so c

Answer 20

Those would be correct :) So that would be our 2nd vector. Now just to make sure we aren't crazy, we can easily check if they're orthogonal. We have the formula
\[\frac{ u*v }{ ||u||||v||}= \cos \theta\] Not sure how to do the actual dot product dot, lol. But if you were to do this operation out, you would end up with 0 = cos(theta), implying theta is pi/2 or 90 degrees and hence theyre orthogonal. So yeah, you were write, but just doing out all the work to show it comes out correct : )