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Question

Decompose the vector 6i^-3j^-6k^ into vectors which are parallel and perpendicular to the vector i^+j^+k^.

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Solution

Let a=6i^-3j^-6k^ and b=i^+j^+k^ and x and y be such thata=x+yy=a-x ... 1Since x is parallel to b,x=t bx=t i^+j^+k^=t i^+t j^+t k^ ...(2)Substituting the values of x and a in (1), we gety=6i^-3j^-6k^-t i^+t j^+t k^ =6-t i^+-3-tj^+-6-t k^ ... 3Since y is perpendicular to b,y. b=06-t i^+-3-tj^+-6-t k^. i^+j^+k^=01 6-t+1-3-t + 1 -6-t =0-3-3t=0t=-1From (2) and (3), we getx=-i^-j^-k^y=7 i^-2j^-5 k^So,a=x+y=-i^-j^-k^+7 i^-2j^-5 k^

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