Question

The position vectors of three consecutive vertices A, B and C of a parallelogram ABCD are $\overrightarrow{r_1},\overrightarrow{r_2}, $ and $\overrightarrow{r_3}, $ respectively. Then the position vector of the fourth vertex D is

• A. $\overrightarrow{r_1}+\overrightarrow{r_2}-\overrightarrow{r_3} $
• B. $\overrightarrow{r_2}+\overrightarrow{r_3}-\overrightarrow{r_1} $
• C. $\overrightarrow{r_3}+\overrightarrow{r_1}-\overrightarrow{r_2} $
• D. none of these.

Answer 1

Answer: (C)

Since the diagonals of a parallelogram bisect each other
$\therefore \frac{\vec{r}_{1}+\vec{r}_{3}}{2}=\frac{\vec{r}_{2}+\vec{r}_{4}}{2}$
$=\vec{r}_{1}+\vec{r}_{3}=\vec{r}_{2}+\vec{r}_{4}$
$\therefore \vec{r}_{4}=\vec{r}_{3}+\vec{r}_{1}-\vec{r}_{2}$, where $D$ is $\left(\vec{r}_{4}\right)$