Calculate the scalar product of the following vectors.
The length of the hypotenuse AB of a right triangle ABC is c. Find the sum $- - \to A B \cdot - - \to A C + - - \to B C \cdot - - \to B A + - - \to C A \cdot - - \to C B .$

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Solution

Since $A B$ is given to be the hypotenuse, $∠ C = 90^{0}$
$⟹ \to C A . \to C B = 0$

$\to A B . \to A C + \to B C . \to B A + \to C A . \to C B = \to A B . \to A C + \to C B . \to A B$
$= \to A B (\to A C + \to C B)$
$= \to A B . \to A B$
$= | \to A B |^{2}$
$= c^{2}$

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