A line L passing through the point P(-2, -1, 7) intersects the line L₁ : x 2 = y-1 2 = z+1 1 at a point Q . If the angle between the lines L and L₁ is ⁻¹ ( 1 3 ) , then the distance PQ is :

Step-by-step solution

Let the point Q on the line L₁ be (2k, 2k+1, k-1) . The direction ratios of the line L (which is the line segment PQ ) are proportional to the vector v = PQ : v = 2k - (-2), 2k+1 - (-1), k-1 - 7 = 2k+2, 2k+2, k-8 The direction vector of the line L₁ is u = 2, 2, 1 . The angle between the lines L and L₁ is given by = 1 3 . Using the dot product formula: = | u v | | u | | v | Calculate the dot product: u v = 2(2k+2) + 2(2k+2) + 1(k-8) = 4k + 4 + 4k + 4 + k - 8 = 9k Calculate the magnitudes: | u | = 2^2 + 2^2 + 1^2 = 9