The value of for which the angle between lines r = + + k +p(2 + +2 k ) and r =(1+q) +(1+q ) +(1+q) k is 2

Step-by-step solution

The direction vector of the first line is v ₁ = 2 i + j + 2 k . The second line is given by r = ( i + j + k ) + q( i + j + k ) . Thus, the direction vector of the second line is v ₂ = i + j + k . Since the angle between the lines is 2 , the lines are perpendicular, which implies that the dot product of their direction vectors must be zero. v ₁ v ₂ = 0 (2 i + j + 2 k ) ( i + j + k ) = 0 (2)(1) + (1)( ) + (2)(1) = 0 2 + + 2 = 0 + 4 = 0 = -4 Answer: -4