Let O be the origin. Let O P → = x i ^ + y j ^ - k ^ and O Q → = - i ^ + 2 j ^ + 3 x k ^ , x , y ∈ R , x > 0 , be such that | P Q → | = 20 and the vector O P → is perpendicular to O Q → . If O R → = 3 i ^ + z j ^ - 7 k ^ , z ∈ R , is coplanar with O P → and O Q → , then the value of x 2 + y 2 + z 2 is equal to

Step-by-step solution

If two vectors a 1 i ^ + b 1 j ^ + c 1 k ^ and a 2 i ^ + b 2 j ^ + c 2 k ^ are perpendicular, then a 1 a 2 + b 1 b 2 + c 1 c 2 = 0 . Given O P → ⊥ O Q → ⇒ - x + 2 y - 3 x = 0 ⇒ y = 2 x       . . . i Also, P Q → = O Q → - O P → ⇒ P Q → = - 1 - x i ^ + 2 - y j ^ + 3 x + 1 k ^ And P Q → = 20 ⇒ - 1 - x 2 + 2 - y 2 + 3 x + 1 2 = 20 ⇒ ( x + 1 ) 2 + ( y - 2 ) 2 + ( 1 + 3 x ) 2 = 20 Put the value from equation i , to get ( x + 1 )