If the line x - 1 2 = y - 2 3 = z - 4 4 intersect the x y and y z plane at points A and B respectively. If the volume of the tetrahedron O A B C is V cubic units (where, O is the origin) and point C is 1,0 , 4 , then the value of 102 V is equal to

Step-by-step solution

On the x y plane, z = 0 ⇒ x - 1 2 = y - 2 3 = - 1 ⇒ x = - 1 , y = - 1 ⇒ Coordinates of point A are - 1 , - 1,0 On the y z plane, x = 0 ⇒ - 1 2 = y - 2 3 = z - 4 4 ⇒ y = 1 2 , z = 2 ⇒ Coordinates of point B are 0 , 1 2 , 2 O A → = - i ^ - j ^ O B → = 1 2 j ^ + 2 k ^ O C → = i ^ + 4 k ^ The volume of the tetrahedron O A B C is V = 1 6 O A →   O B →   O C → = 1 6 - 1 - 1 0 0 1 2 2 1 0 4 = 1 6 - 1 2 + 1 - 2 = 4 6 = 2 3 cubic units Hence,