If 7 j + 10 k , - i + 6 j + 6 k and -4 i + 9 j + 6 k are the position vectors of the vertices A, B and C repectively of ABC . Then the position vector of the point where the bisector of the angle A meets side BC is

Step-by-step solution

Let the position vectors of the vertices be a = 7 j + 10 k , b = - i + 6 j + 6 k , and c = -4 i + 9 j + 6 k . First, we find the vectors AB and AC and their magnitudes: AB = b - a = (- i + 6 j + 6 k ) - (7 j + 10 k ) = - i - j - 4 k | AB | = (-1)^2 + (-1)^2 + (-4)^2 = 1 + 1 + 16 = 18 = 3 2 AC = c - a = (-4 i + 9 j + 6 k ) - (7 j + 10 k ) = -4 i + 2 j - 4 k | AC | = (-4)^2 + 2^2 + (-4)^2 = 16 + 4 + 16 = 36 = 6 Let D be the point where the bisector of angle A meets the side BC . By the angle bisector theorem, D divid