a = i - j , b = j - k , c = k - i then a unit vector d such that a d =0=[ b c d ] is

Step-by-step solution

Let d =x i +y j +z k be a unit vector orthogonal to a = i - j and coplanar with b = j - k , c = k - i . The orthogonality condition a d =0 gives x-y=0 , so x=y . The coplanarity of b , c , d is equivalent to [ b c d ]=0 , which implies d is orthogonal to b c . Compute b c = vmatrix i & j & k 0 & 1 & -1 -1 & 0 & 1 vmatrix = i (1 1-(-1) 0)- j (0 1-(-1) (-1))+ k (0 0-1 (-1))= i + j + k . Then d ( b c )=x+y+z=0 . With x=y , we get 2x+z=0 , so z=-2x . Thus d =x( i + j -2 k ) . The unit vector condition requires |x| 1^2+