Given, ( a =2 i + j + k , b = i +2 j - k ) and (a ) unit vector ( c ) are coplanar. If ( c ) is perpendicular to a, then ( c = )

Step-by-step solution

Given, ( array r a =2 i + j + k b = i +2 j - k array ) Let ( c =x i +y j +z k ) ( array lrl & & c a =0 & (x i +y j +z k ) (2 i + j + k )=0 & 2 x+y+z=0 (i) array ) Since, ( a , b ) and ( c ) are coplanar. ( array lc & [ a b c ]=0 & [ array ccc x & y & z 2 & 1 & 1 1 & 2 & -1 array ]=0 & x(-1-2)-y(-2-1)+z(4-1)=0 & -3 x+3 y+3 z=0 & -x+y+z=0 & x=y+z (ii) array ) By solving Eqs. (i) and (ii), we get ( gathered x=0, y=-c x^2+y^2+z^2=1 y= 1 2 c = 1 2 (- j + k ) gathered )