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Vector Algebra

Question
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If straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space not equal to space 0 with rightwards arrow on top comma      then prove that  straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top space equals space straight m space straight b with rightwards arrow on top where m is a scalar.

Solution
straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top cross times straight c with rightwards arrow on top space space space space space rightwards double arrow space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight c with rightwards arrow on top space cross times space straight b with rightwards arrow on top space space space space space rightwards double arrow space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space space space left parenthesis straight a with rightwards arrow on top cross times straight c with rightwards arrow on top right parenthesis space cross times space straight b with rightwards arrow on top space equals 0 with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis because space space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space space equals space straight b with rightwards arrow on top space cross times straight c with rightwards arrow on top space not equal to 0 with rightwards arrow on top space therefore space space space space none space of space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space space is space straight a space zero space vector therefore space space space from space left parenthesis 1 right parenthesis comma space it space follows space that space straight a with rightwards arrow on top plus straight c with rightwards arrow on top space is space parallel space to space straight b with rightwards arrow on top therefore space space space space straight a with rightwards arrow on top plus straight c with rightwards arrow on top space equals space straight m space straight b with rightwards arrow on top space space where space straight m space is space some space scalar. space space space space space space space space space space space space space space space space space space space space space space space space space space

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