Question

$A,B,C,D,E,$ and $F$ in that order, are the vertices of a regular hexagon with centre origin. If the position vectors of the vertices $A$ and $B$ are respectively, $4\hat{i}+3\hat{j}-\hat{k}$ and $-3\hat{i}+\hat{j}+\hat{k}$, then $DE$ is equal to?

• A. $7\hat{i}+2\hat{j}-2\hat{k}$
• B. $-7\hat{i}-2\hat{j}+2\hat{k}$
• C. $3\hat{i}-\hat{j}-\hat{k}$
• D. $-4\hat{i}-3\hat{j}+2\hat{k}$

Answer 1

The correct option is A

$7\hat{i}+2\hat{j}-2\hat{k}$

Explanation for the correct option:

Find the value of $\overrightarrow{DE}$:

Let $O$ is the center origin of the regular hexagon

$\because A,B,C,D,E,F$ are the vertices of given regular hexagon in that order

$\therefore$$\overrightarrow{AB}$ and $\overrightarrow{DE}$ will be with equal magnitude but in opposite direction.

$\Rightarrow$$\overrightarrow{AB}=-\overrightarrow{DE}$

Now,

$\because \overrightarrow{AB}=\overrightarrow{OB}–\overrightarrow{OA}$

$\overrightarrow{OB}=-3\hat{i}+\hat{j}+\hat{k}$

$\overrightarrow{OA}=4\hat{i}+3\hat{j}-\hat{k}$

Put the value $\overrightarrow{OB}$ and $\overrightarrow{OA}$ in $\overrightarrow{AB}$,

$\Rightarrow$$\overrightarrow{AB}=\left(-3\hat{i}+\hat{j}+\hat{k}\right)-\left(4\hat{i}+3\hat{j}-\hat{k}\right)$

$\Rightarrow$$\overrightarrow{AB}=-7\hat{i}-2\hat{j}+2\hat{k}$

$\therefore \overrightarrow{DE}=-\overrightarrow{DE}=7\hat{i}+2\hat{j}-2\hat{k}$

Hence, option ‘A’ is correct.