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Vector Algebra

Question
CBSEENMA12034241

If   straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space show that 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 equals straight c with rightwards arrow on top cross times straight a with rightwards arrow on top. Interpret the result geometrically.

Solution

Now, straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c 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 rightwards double arrow space space space space straight a with rightwards arrow on top plus straight b with rightwards arrow on top space equals space minus straight c with rightwards arrow on top space space space space rightwards double arrow space space space straight a with rightwards arrow on top space cross times left parenthesis straight a with rightwards arrow on top cross times straight b with rightwards arrow on top right parenthesis space equals space minus straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top
rightwards double arrow space space space space straight a with rightwards arrow on top cross times straight a with rightwards arrow on top plus 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 cross times straight a with rightwards arrow on top space space rightwards double arrow space space space 0 with rightwards arrow on top plus 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 a with rightwards arrow on top space space space space space rightwards double arrow space 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 a with rightwards arrow on top
                                                                                                                                 ...(1)
Similarly,   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                                                                                  ...(2)
From (1) and (2), we get, 
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 space cross times space straight c with rightwards arrow on top space equals straight c with rightwards arrow on top cross times space straight a with rightwards arrow on top
Let  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 equals space straight c with rightwards arrow on top cross times straight a with rightwards arrow on top space equals straight p with rightwards arrow on top
therefore space space space straight p with rightwards arrow on top space is space perpendicular space to space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space and space straight c with rightwards arrow on top
rightwards double arrow space space straight p with rightwards arrow on top space is space perpendicular space to space the space plane 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
rightwards double arrow space 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 are space coplanar.