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Math Help - vector proof

  1. #1
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    vector proof

    (1)Given two vectors , a and b where a and b are both not 0 , show that if |a+b|=|a-b| , then a and b are perpendicular to each other .

    (2) if a and b are perpendicular to each other , show that

    |a+b|=|a-b|= \sqrt{|<b>a</b>|^2+|<b>b</b>|^2}
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  2. #2
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    Quote Originally Posted by hooke View Post
    (1)Given two vectors , a and b where a and b are both not 0 , show that if |a+b|=|a-b| , then a and b are perpendicular to each other .

    (2)...
    Square both sides of the equation to get rid of the absolute values:

    |\vec a + \vec b | = |\vec a - \vec b | ~\implies~(\vec a)^2 + 2 \vec a \vec b + (\vec b)^2 = (\vec a)^2 - 2 \vec a \vec b + (\vec b)^2

    Collect like terms:

    2 \vec a \vec b = -2 \vec a \vec b~\implies~4 \vec a \vec b = 0

    Thus \vec a \vec b = 0 which means \vec a \perp \vec b
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