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Vector Algebra

Question
CBSEENMA12034246
Wired Faculty App

If straight a with rightwards arrow on top not equal to 0 with rightwards arrow on top comma space space do comma space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space and space space straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top space equals straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space together space imply space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top ?

Solution

We have,    straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top. space straight c with rightwards arrow on top
     therefore space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 0 space space space space rightwards double arrow space space space space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top right parenthesis space equals space 0 therefore space space space space there space are space three space possibilities colon
(i) straight a with rightwards arrow on top space equals space 0 degree           left parenthesis ii right parenthesis space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals 0 with rightwards arrow on top                (iii) straight a with rightwards arrow on top is perpendicular to straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top
Again,              straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top
therefore space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space cross times straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space space space rightwards double arrow space space straight a with rightwards arrow on top 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 therefore space space space again space there space are space three space possibilities colon left parenthesis straight i right parenthesis space space straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space space space left parenthesis ii right parenthesis space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space space space left parenthesis iii right parenthesis space straight a with rightwards arrow on top space is space parallel space to space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top Now comma space straight a with rightwards arrow on top space is space given space to space be space non minus zero space vector therefore space space space space we space have space the space following space possibilities space left colon left parenthesis 1 right parenthesis space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top left parenthesis 2 right parenthesis space straight a with rightwards arrow on top space is space perpendicular space to space straight b with rightwards arrow on top minus straight c with rightwards arrow on top space space and space straight a with rightwards arrow on top space is space parllel space to space straight b with rightwards arrow on top minus straight c with rightwards arrow on top comma space which space is space absurd. therefore space space space space space space only space possibility space left space is space straight b with rightwards arrow on top minus straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top space space or space space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top. space space space space

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