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Math Help - Vectors Application 2

  1. #1
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    Vectors Application 2

    If vector u and vector v are non-zero vectors, but Proj(u onto v) = vector 0, what conclusion can be drawn? If Proj(u onto v)= vector 0, does it follow that Proj(v onto u)=vector 0? Explain

    Ok, so I am having trouble trying to explain this question. This is what I came up with so far, hopefully someone can add on or help answer this question better. Thanks for your help in advance.

    Part A
    when Proj(u onto v)=vector 0, that means that the projection vector is at the orgin, doesn't it?

    Part B
    I know that they are not equal to each other.. but I am not sure how to prove this.
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  2. #2
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    It means simply they are perpindicular

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

    so u*v = 0 u and v are perpindicular
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  3. #3
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    Quote Originally Posted by skeske1234 View Post
    If vector u and vector v are non-zero vectors, but Proj(u onto v) = vector 0, what conclusion can be drawn? If Proj(u onto v)= vector 0, does it follow that Proj(v onto u)=vector 0? Explain

    Ok, so I am having trouble trying to explain this question. This is what I came up with so far, hopefully someone can add on or help answer this question better. Thanks for your help in advance.

    Part A
    when Proj(u onto v)=vector 0, that means that the projection vector is at the orgin, doesn't it?
    No, that would be the 0 vector and you are told that u and v are non-zero vectors. Since u and v are non-zero vectors, |u| and |v| are non-zero so you can write the projection of u on v as \frac{u\cdot v}{|u||v|}v. In order that that be the 0 vector, u\cdot v must be 0 so the vectors must be perpendicular.

    Part B
    I know that they are not equal to each other.. but I am not sure how to prove this.
    While, in general, the "projection of u on v" is not the same as the "projection of v on u", being perpendicular to one another is "dual": if u is perpendicular to v, then v is perpendicular to u. If the projection of u on v is 0, then, by A, one of three things must be true:
    a) u= 0
    b) v= 0
    c) u is perpendicular to v.

    In any of those, it follows that the projection of v on u is also 0.
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