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Vector Algebra

Question
CBSEENMA12034259

If 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 are three vectors such that 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 comma space space space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space show that the three vectors straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top are orthogonal in pairs and open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals 1.

Solution
because space space space space 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
therefore space space straight c with rightwards arrow on top space is space vector space which space is space perpendicular space to space straight a with rightwards arrow on top space as space well space as space straight b with rightwards arrow on top.
Again comma space straight b with rightwards arrow on top space cross times straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top
therefore space space space straight a with rightwards arrow on top space is space perpendicular space to space straight b with rightwards arrow on top space as space well space as space straight c with rightwards arrow on top.
therefore space space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top space are space mutually space orthogonal
Again comma space space space 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
rightwards double arrow space space space space space space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar
rightwards double arrow space space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar 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 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 space space space
                                                                                 open square brackets because space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space are space perpendicular close square brackets
Again,  straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top
rightwards double arrow space space space space space open vertical bar straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top close vertical bar space equals straight a with rightwards arrow on top                          rightwards double arrow space space space space space space space space space space open vertical bar straight b with rightwards arrow on top close vertical bar space space open vertical bar straight c with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top close vertical bar                 ...(2)
                                                                                   open square brackets because space space straight b with rightwards arrow on top comma space space straight c with rightwards arrow on top space are space perpendicular close square brackets
From (1) and (2),
                      open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight c with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar space space space space space space space rightwards double arrow space space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals space 1 space space space space space space space space space space space rightwards double arrow space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 1
therefore space space space space from space left parenthesis 1 right parenthesis comma space space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar