Passage: P(-2,-3,5) , Q(4,-1,5) , R(6,-4,8) and S(2,-6,10) are four points. Question: What is (PQ^2+2QS^2-2PR^2) equal to ?

Step-by-step solution

Given points are P(-2,-3,5) , Q(4,-1,5) , R(6,-4,8) and S(2,-6,10) . Using the distance formula, we find the squares of the distances between the required points: PQ^2 = (4 - (-2))^2 + (-1 - (-3))^2 + (5 - 5)^2 = 6^2 + 2^2 + 0^2 = 36 + 4 = 40 QS^2 = (2 - 4)^2 + (-6 - (-1))^2 + (10 - 5)^2 = (-2)^2 + (-5)^2 + 5^2 = 4 + 25 + 25 = 54 PR^2 = (6 - (-2))^2 + (-4 - (-3))^2 + (8 - 5)^2 = 8^2 + (-1)^2 + 3^2 = 64 + 1 + 9 = 74 Substituting these values into the given expression: PQ^2 + 2QS^2 - 2PR^2 = 40 + 2(54) - 2(74) = 40 +