Question

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

• A. $r_1+r_2-r_3$
• B. $r_2+r_3-r_1$
• C. $r_3+r_1-r_2$
• D. None of these

Answer 1

The correct option is C

$r_3+r_1-r_2$

Explanation for the correct option:

Find the position vector of the fourth point

In the question, The position vectors of three consecutive vertices $A,B$ and $C$ of a parallelogram $ABCD$ are $r_1,r_2$ and $r_3$ respectively are given.

Assume that, the position vector of the fourth point $D$ is $r_4$.

We know that the diagonals of a parallelogram bisect each other.

Since the mid-point of the two position vectors $x$ and $y$ is given by $\frac{x+y}{2}$.

So,

$$\begin{gathered} \frac{r_1+r_3}{2}=\frac{r_2+r_4}{2} \\ \Rightarrow r_1+r_3=r_2+r_4 \\ \Rightarrow r_4=r_1+r_3-r_2 \end{gathered}$$

Therefore, the position vector of the fourth vertex $D$ is $r_3+r_1-r_2$.

Hence, option $C$, $r_3+r_1-r_2$ is the correct answer.