Lines r = a + b and r = b + a intersect at point (2, 4, -4) . If | a - b | = 4 , then a b =

Step-by-step solution

Let the point of intersection be p = 2 i + 4 j - 4 k . Since the lines intersect at p , there exist scalars and such that: a + b = b + a (1 - ) a + ( - 1) b = 0 Assuming a and b are non-collinear, we get 1 - = 0 and - 1 = 0 , which gives = 1 and = 1 . Thus, the position vector of the point of intersection is: p = a + b The magnitude squared of p is: | a + b |^2 = 2^2 + 4^2 + (-4)^2 = 4 + 16 + 16 = 36 We are given | a - b | = 4 , so: | a - b |^2 = 16 Using the vector identity: | a + b |^2 - | a - b |^2 = 4 a b Subst