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Math Help - Prove that these two vectors are orthogonal

  1. #1
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    Prove that these two vectors are orthogonal

    ||b||a + ||a||b and ||b||a - ||a||b

    Here's my attempt:

    Since I wish to show that these vectors are orthogonal, I will attempt to show that the dot product must be 0.

    (||b||a + ||a||b) * (||b||a - ||a||b)

    = ||b||(a * a) + (||b||a * ||a||b) + (||a||b * ||b||a) + ||a||(b * -b)

    = ||b||||a||^2 + ||b||^2||a||^2 - ||b||^2||a||^2 + ||a||||b||^2(-1)

    = ||b||||a||^2 - ||a||||b||^2

    But I see nothing to suggest that this is 0.

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  2. #2
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    Re: Prove that these two vectors are orthogonal

    Nevermind, I made a careless error and that last line should have all magnitudes squared.
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  3. #3
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    Re: Prove that these two vectors are orthogonal

    Quote Originally Posted by OneMileCrash View Post
    ||b||a + ||a||b and ||b||a - ||a||b

    Here's my attempt:

    Since I wish to show that these vectors are orthogonal, I will attempt to show that the dot product must be 0.

    (||b||a + ||a||b) * (||b||a - ||a||b)

    = ||b||(a * a) + (||b||a * ||a||b) + (||a||b * ||b||a) + ||a||(b * -b)

    = ||b||||a||^2 + ||b||^2||a||^2 - ||b||^2||a||^2 + ||a||||b||^2(-1)
    Line 3 should read (after canceling the middle terms)
    = ||b||^2 * ||a||^2 - ||a||^2 * ||b||^2

    -Dan
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