Let (a=i+j+k, b=i-j+k ) and (c=i-j-k ) be three vectors. A vector (v ) in the plane of a and (b ), whose projection on (c ) is (1 / 3 ), is given by

Step-by-step solution

Given that, [ array l a = i +2 j + k (1) b = i - j + k (2) C = i + j - k (3) array ] A vector in the plane of ( a ) and ( b ) can be written as, [ array l r =m a +n b =m( i +2 j + k )+n( i - j + k ) r =(m+n) i +(2 m-n) j +(m+n) k array ] So, projection on ( C ) by ( r ) is given by [ array l r C | C | = (m+n)+(2 m-n)-(m+n) 1+1+1 2 m-n 3 = 1 3 array ] As use can see from Eq. (4), the ( ^ i ) component and the ( k ) component must be equal and ( j ) can be either ( -1 ) or ( 1 . ) Therefore, option (4) is correct, th