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Math Help - Vector analysis

  1. #1
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    Vector analysis

    Hi, I am having trouble with this problem :

    Let A and B be 2 vectors. Express the vector B as the sum of a vector C, that is parallel to A, and of a vector D that is perpendicular to A. Note : This is in 3-D space.

    I don't even know how to start.

    Thanks in advance
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  2. #2
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by yuki267 View Post
    Hi, I am having trouble with this problem :

    Let A and B be 2 vectors. Express the vector B as the sum of a vector C, that is parallel to A, and of a vector D that is perpendicular to A. Note : This is in 3-D space.

    I don't even know how to start.

    Thanks in advance
    Maybe you could start by writing the components of A, for example A=(a_1,a_2,a_3). The same for B: B=(b_1,b_2,b_3).
    Do the same for C and D.
    Now, C is parallel to A. It means that their cross product is null. (see Cross Product -- from Wolfram MathWorld). So do the cross product between those 2 vectors and equal it to 0.
    D is orthogonal to A, it means that their dot product is null.
    Can you take it from there?
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  3. #3
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    Quote Originally Posted by yuki267 View Post
    Let A and B be 2 vectors. Express the vector B as the sum of a vector C, that is parallel to A, and of a vector D that is perpendicular to A. Note : This is in 3-D space.
    Try these.
    C=\frac{A\cdot B}{A\cdot A}A~~\&~~D=B-C
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  4. #4
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    What they area asking you to do is to find the "parallel projection" and "orthogonal projection" of A on B. Plato gave you the formulas for that.
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  5. #5
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    What they area asking you to do is to find the "parallel projection" and "orthogonal projection" of A on B. Plato gave you the formulas for that.
    Oh I missed this. I think my method still works but I agree it's much longer than Plato's one.
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