# The Vector AB= 3i + 4k and  AC = 5i – 2j + 4k are the sides of a triangle ABC. The length of the median through is(A)  18½    (B)  72½    (C)   33½    (D)  288½

Piyush Kumar Behera
417 Points
6 years ago
A little easy way of approach,
Take A as the origin(0,0)
so Vector AB= 3i + 4k and  AC = 5i – 2j + 4k will become position vector.
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=$\sqrt{(4-0)^2+(-1-0)^2+(4-0)^2}$
=$\sqrt{33}$
I have put my best to explain.If you are unable to get please do write I will try to explain again.And please If my answer is correct do approve my answer!!

3 years ago
Dear student,

This question would become easy to solve if you notice the similarity between parralelogram law of vector addition and the fact that the diagonal represents the resultant vector and also the diagonals bisects each other.
Let us assume the resultant of vector AC and AB is vector AD. As we know diagonals bisect each other. Hence half of the resultant vector AD will be the median for side BC
Hence, Vector AD = (AC + AB)/2 = (5i – 2j + 4k + 3i + 4k)/2
= (8i – 2j + 8k)/2
= 4i – j + 4k
Hence the length of median = |AD| = (42 + 12 + 42)½
= 33½
Hence C) is the correct option.

Hope it helps.
Thanks and regards,
Kushagra