The line of shortest distance between the lines L₁ : x-1 2 = y+1 1 = z-2 -2 and L₂ intersects L₁ at the point P(3, 0, 0) and L₂ at the point Q . If the direction ratios of L₂ are (1, 2, 2) , the shortest distance between L₁ and L₂ is 6 , and the x -coordinate of Q is positive, then the value of OQ^2 (where O is the origin) is equal to ________.

Step-by-step solution

The line of shortest distance between L₁ and L₂ is perpendicular to both lines. Let d₁ = 2 i + j - 2 k and d₂ = i + 2 j + 2 k . The direction vector of the line of shortest distance is: n = d₁ d₂ = vmatrix i & j & k 2 & 1 & -2 1 & 2 & 2 vmatrix = 6 i - 6 j + 3 k This can be simplified to 2 i - 2 j + k . The line of shortest distance passes through P(3, 0, 0) and has direction ratios (2, -2, 1) . The parametric coordinates of any point on this line are (3+2t, -2t, t) . Since Q lies on this line, Q = (3+2t, -2t, t) f