a =2 i - j , b =2 j - k , c =2 k - i are three vectors and d is a unit vector perpendicular to c . If a , b , d are coplanar vectors, then | d b |=

Step-by-step solution

a =2 i - j , b =2 j - k , c =- i +2 k Let d = p i +q j +r k p^2+q^2+r^2 , let d is ^ r to c d c =0 -p+2 r=0 p=2 r ...(i) a , b , d are coplanar, means aligned & a ( b d )=0 or [a b d]=0 & | array ccc 2 & -1 & 0 0 & 2 & -1 p p^2+q^2+r^2 & q p^2+q^2+r^2 & r p^2+q^2+r^2 array |=0 aligned aligned & 2(2 r+q)+p=0 q=-3 r...(ii) & d b = |2 q-r| p^2+q^2+r^2 = |-6 r-r| 4 r^2+9 r^2+r^2 & = 7 r 14 r^2 = 49 14 = 7 2 . aligned