A unit vector coplanar with i + j +3 k and i +3 j + k and perpendicular to i + j + k is

Step-by-step solution

Let the unit vector be aligned & r =x i +y j +z k & and a = i + j +3 k , b = i +3 j + k & and c = i + j + k aligned Given, [ r , a b ]=0 , i.e., coplanar. array ccc & | array ccc x & y & z 1 & 1 & 3 1 & 3 & 1 array |=0 & x(1-9)-y(1-3)+z(3-1)=0 & -8 x+2 y+2 x=0 & -4 x+y+z=0 array and r c =0 , i.e., perpendicular array cc & (x i +y j +z k ) ( i + j + k )=0 & x+y+z=0 array On solving Eqs. (i) and (ii), we get aligned 5 y+5 z & =0 y & =-z aligned r is a unit vector. array ll & | r |=1= x^2+y^2+z^2 & x^2+y^2+z^2=1 & x^2