The length of the perpendicular from the point P(1,-1,2) to the given line x+1 2 = y-2 -3 = z+2 4 is

Step-by-step solution

Let the line be L: x+1 2 = y-2 -3 = z+2 4 = . Any point Q on the line L is given by Q(2 - 1, -3 + 2, 4 - 2) . The vector PQ is given by (2 - 1 - 1, -3 + 2 - (-1), 4 - 2 - 2) = (2 - 2, -3 + 3, 4 - 4) . Since PQ is perpendicular to the direction vector of the line v = (2, -3, 4) , their dot product must be zero: (2 - 2)(2) + (-3 + 3)(-3) + (4 - 4)(4) = 0 4 - 4 + 9 - 9 + 16 - 16 = 0 29 - 29 = 0 = 1 . Substituting = 1 into the coordinates of Q , we get Q(2(1) - 1, -3(1) + 2, 4(1) - 2) = Q(1, -1, 2) . Since the point P(