Q. $A,B,C,D,E,F\$in that order, are the vertices of a regular hexagon with center origin. If the position vectors $A\$and $B$ are respectively, $4\ \widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$ and $- 3\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}},$ then $\overrightarrow{\mathbf{DE}}$ is equal to
A) $7\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}}$
B) $- 7\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
C) $3\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
D) $- 4\widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
Explanation
(a)
Given $\overrightarrow{\mathbf{OA}} = 4\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
$\overrightarrow{\mathbf{OB}} = - 3\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$
$\therefore\ \overrightarrow{\mathbf{AB}} = \overrightarrow{\mathbf{OB}} - \overrightarrow{\mathbf{OA}} = - 7\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
$\therefore\ \overrightarrow{\mathbf{DE}} = - \overrightarrow{\mathbf{AB}} = 7\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.