Let a line L passing through the point (1, 1, 1) be perpendicular to both the vectors 2 i + 2 j + k and i + 2 j + 2 k . If P(a, b, c) is the foot of perpendicular from the origin on the line L , then the value of 34(a + b + c) is :

Step-by-step solution

The direction vector of the line L is given by the cross product of the two vectors to which it is perpendicular: d = (2 i + 2 j + k ) ( i + 2 j + 2 k ) = 2 i - 3 j + 2 k The equation of the line L passing through (1, 1, 1) is: x - 1 2 = y - 1 -3 = z - 1 2 = Any point P on the line L has coordinates (2 + 1, -3 + 1, 2 + 1) . Since P(a, b, c) is the foot of the perpendicular from the origin (0, 0, 0) to the line L , the vector OP is perpendicular to the direction vector d of the line. Thus, their dot product is zero: