Problem: For vectors u and v, we have p=proj_v(u). If norm(u)=11 and norm(p)=6, find p * u.

beefypy

beefypy

Answered question

2022-10-24

Problem:For vectors u and v, we have p = proj v ( u ). If u = 11 and p = 6, find p u
I don't understand how to do this problem because, to me, it seems like there isn't enough information, but I know that there is. I know that any vector and its projection either form the hypotenuse and a leg of a right triangle, or that they are equal. So I can figure out the cosine of the angle between u and p. But, I don't know how exactly I would do that?

Answer & Explanation

vacchetta7k

vacchetta7k

Beginner2022-10-25Added 6 answers

Remember that the formula for vector projection is
p r o j v ( u ) = p = ( u v v 2 ) v
Supposing θ is the angle between u and v, this can be written as
p = ( u v cos ( θ ) v 2 ) v = ( u cos ( θ ) v ) v
Therefore
p = ( u cos ( θ ) v ) v = u cos ( θ )
Now, since v and p r o j v ( u ) point in the same direction, the angle between u and p is also θ, therefore
u p = u p cos ( θ ) = p 2 .
independanteng

independanteng

Beginner2022-10-26Added 4 answers

Since the length of the projection of u on v is 6, and the length of u is 11, then if the angle between u and its protection p on v is ϕ , then we must have
6 = 11 cos ϕ .
Now since p u = | p | | u | cos ϕ , and we know | p | = 6 , | u | = 11 and cos ϕ = 6 / 11 , can you now finish it off?

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