Question

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Answer 1

Let `vec"a", vec"b" and vec"c"` are the position vectors of the vertices A, B and C respectively.

Then we know that the position vector of the centroid O of the triangle is `(vec"a" + vec"b" + vec"c")/3`

Therefore sum of the three vectors `vec"OA", vec"OB" and vec"OC",` is

`vec"OA" +  vec"OB" + vec"OC" = vec"a" - ((vec"a" + vec"b" + vec"c")/3) + vec"b" - ((vec"a" + vec"b" + vec"c")/3) + vec"c" - ((vec"a" + vec"b" + vec"c")/3)`

`= (vec"a" + vec"b" + vec"c") - 3((vec"a" + vec"b" + vec"c")/3)`

`= vec0`

Hence, Sum of the three vectors determined by the medians of a triangle directed from the vertices is zero.