The vectors AB = 3^i + 4^k and AC = 5^i - 2^j + 4^k are the sides of a triangle ABC, then the length of the median through A is:

Take A as the origin (0,0) so AB = 3^i + 4^k and AC = 5^i - 2^j + 4^k will become position vectors. So coordinate of B is (3,0,4) and C is (5,-2,4). The mid point of BC can be easily found by midpoint formula of two points as D(4,-1,4). so length of median through A is AD = √(4-0)² + (-1-0)² + (4-0)² = √33