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Vector Algebra

Question
CBSEENMA12034253
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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 c with rightwards arrow on top space cross times space straight d with rightwards arrow on top space and space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top space cross times space straight d with rightwards arrow on top comma space show space that space straight a with rightwards arrow on top space minus space straight d with rightwards arrow on top is parallel to straight b with rightwards arrow on top minus straight c with rightwards arrow on top space provided space straight a with rightwards arrow on top space not equal to space straight d with rightwards arrow on top space and space straight b with rightwards arrow on top not equal to straight c with rightwards arrow on top.

Solution
open table attributes columnalign right end attributes row cell We space have space straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight d with rightwards arrow on top end cell row cell and space straight a with rightwards arrow on top cross times space straight c with rightwards arrow on top space equals straight b with rightwards arrow on top space cross times space straight d with rightwards arrow on top end cell end table close curly brackets 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
straight a with rightwards arrow on top space minus space straight d with rightwards arrow on top will be parallel to straight b with rightwards arrow on top minus straight c with rightwards arrow on top

if left parenthesis straight a with rightwards arrow on top space minus space straight d with rightwards arrow on top right parenthesis space cross times space left parenthesis straight b with rightwards arrow on top minus straight c with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top space space space straight i. straight e. comma space if space straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space minus space straight d with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight d with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top
straight i. straight e. comma space if space left parenthesis straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight d with rightwards arrow on top space cross times space straight c with rightwards arrow on top right parenthesis space minus space left parenthesis straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space plus space straight d with rightwards arrow on top space cross times space straight b with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top straight i. straight e. comma space if space left parenthesis straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space cross times space straight d with rightwards arrow on top right parenthesis space minus space left parenthesis straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space minus straight b with rightwards arrow on top space cross times space straight d with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top straight i. straight e. comma space if space 0 with rightwards arrow on top space minus space 0 with rightwards arrow on top space equals space 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 open square brackets because space of space left parenthesis 1 right parenthesis close square brackets straight i. straight e. comma space if space 0 with rightwards arrow on top space equals space 0 with rightwards arrow on top space which space is space true.
Hence the result. 

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