Prove that for any 2 vectors, U and V, we get ||U+V||^2 + ||U-V||^2 = 2(||U||^2 + ||V||^2). I have proven this with simple algebra. But what geometric fact does it prove?
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addle_brains
ANSWERS: 1
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This can be proven using the law of cosines. |U+V|^2 = |U|^2 + |V|^2 +2*|U|*|V|*cosß |U-V|^2 = |U|^2 + |V|^2 -2*|U|*|V|*cosß Now just add the two to get your equality. This means that in a random parallelogram, the sum of the squares of the diagonals is the sum of the squares of the sides. This is the parallelogram law. Further information: http://en.wikipedia.org/wiki/Law_of_Cosines
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