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Applications of Dot Product

A, B, C, D ar...

Question

$A, B, C, D$ are any four points, prove that $\to A B . \to C D + \to B C . \to A D + \to C A . \to B D = 0$

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Solution

According to question,
$\begin{matrix} - - \to A D = - - \to A B + - - \to B C + - - \to C D = \to a + \to b + \to c - - \to A C = - - \to A B + - - \to B C = \to a + \to b - - \to B D = - - \to B C + - - \to C D = \to b + \to c H e n c e, - - \to C A = - - - \to A C - - \to C A = - (\to a + \to b) N o w, f r o m e q u : \Rightarrow \to a . \to c + \to b . (\to a + \to b + \to c) - (\to a + \to b) . (\to c + \to b) \Rightarrow \to a . \to c + \to b . \to a + \to b . \to b + \to b . \to c - (\to a . \to b + \to a . \to c + \to b . \to b + \to b . \to c) \Rightarrow \to a . \to c + \to b . \to a + \to b . \to b + \to b . \to c - \to a . \to b - \to a . \to c - \to b . \to b - \to b . \to c = 0 \end{matrix}$
So that equation is proved.

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