Let , and be the position vectors of the vertices , and of a quadrilateral taken with…

MATHEMATICS · Subjective · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

Let

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

, and

\mathbf{c}

be the position vectors of the vertices

A,\,B

, and

C

of a quadrilateral

ABCD

taken with respect to an origin

O

. Suppose that

\mathbf{d}

is the position vector of the fourth vertex

D

such that

\mathbf{a}+\mathbf{c}=\mathbf{b}+\mathbf{d}

. (a) Prove that

ABCD

is a parallelogram. (b) Show that the diagonals

AC

and

BD

bisect each other. (c) If

\mathbf{a}=(1,2,3)

\mathbf{b}=(4,-1,0)

, and

\mathbf{c}=(-2,5,6)

, find the coordinates of

D

and verify the result obtained in part (b) using components.