Let be a triangle whose position vectors with respect to the origin are and respectively.

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let

ABC

be a triangle whose position vectors with respect to the origin are

\mathbf{a},\,\mathbf{b}

and

\mathbf{c}

respectively. (i) Find, in vector form, the medians through vertices

A

B

and

C

. (ii) Show that the three medians are concurrent, and obtain the position vector of their common point (centroid) in terms of

\mathbf{a},\,\mathbf{b},\,\mathbf{c}

. (iii) Prove that the vector sum of the three medians of a triangle is the zero vector.