Question

Suppose that a parallelogram has vertices at 0, u, v, and u+v. Show that its diagonals have directions u+v and u−v

Answer 1

Let OACB be the parallelogram with point O at origin as shown in figure below:

u+v u-v 0

Given that the points A, B, have the position vectors u and v respectively.

Then from the parallelogram in triangle ABC, we have

${\color{Yellow} .}\; \; \; \; \; \; \; \; \; \; {\color{Blue} \overrightarrow{OC}=\overrightarrow{OA}+\overrightarrow{AC}}\; \; \; \; \; \; \; \; \; \; [By\; triangle\; law\: of\: vector\: addition]$

${\color{Yellow} .}\; \; \; \; \; \; \; \; \; \; {\color{Blue} \; \; \; \; \; \; \; =\overrightarrow{u}+\overrightarrow{v}}$

Similarly, from triangle OBA, we have

${\color{Yellow} .}\; \; \; \; \; \; \; \; \; \; {\color{Blue} \overrightarrow{OA}=\overrightarrow{OB}+\overrightarrow{BA}}\; \; \; \; \; \; \; \; \; \; [By\; triangle\; law\: of\: vector\: addition]$

${\color{Yellow} .}\; \; \; \; \; \; \; \; \; \; {\color{Blue} \overrightarrow{u}=\overrightarrow{v}+\overrightarrow{BA}}$

${\color{Yellow} .}\; \; \; \; \; \; \; \; \; \; {\color{Blue} \overrightarrow{BA}=\overrightarrow{u}-\overrightarrow{v}}$

Thus, the diagonal vectors OC and BA are u + v and u - v respectively.