The three points A(2, 4, 3), B(4, a, 9) and C(10, -1, 7) form a right-angled triangle with B = 90^ , then the value of "a" is

Step-by-step solution

The given points are A(2, 4, 3) , B(4, a, 9) , and C(10, -1, 7) . The direction ratios of the line segment AB are (2 - 4, 4 - a, 3 - 9) = (-2, 4 - a, -6) . The direction ratios of the line segment BC are (10 - 4, -1 - a, 7 - 9) = (6, -1 - a, -2) . Since B = 90^ , the line segments AB and BC are perpendicular to each other. Therefore, the dot product of their direction vectors is zero. (-2)(6) + (4 - a)(-1 - a) + (-6)(-2) = 0 -12 - (4 - a)(1 + a) + 12 = 0 (a - 4)(a + 1) = 0 a = 4 or a = -1 Answer: -1 or 4