Q. Consider points $A,B,C$ and $D$ with position vectors $7\widehat{\dot{\mathbf{i}}} - 4\widehat{\dot{\mathbf{j}}} + 7\widehat{\mathbf{k}},\widehat{\dot{\mathbf{i}}} - 6\widehat{\dot{\mathbf{j}}} + 10\widehat{\mathbf{k}}$, $- \widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}}$ and $5\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 5\widehat{\mathbf{k}}$ respectively. Then, $ABCD$ is a
A) Square
B) Rhombus
C) Rectangle
D) None of these
Explanation
(d)
$\left| \overrightarrow{\mathbf{AB}} \right| = \sqrt{{(7 - 1)}^{2} + {( - 4 + 6)}^{2} + {(7 - 10)}^{2}} = 7$
$\left| \overrightarrow{\mathbf{BC}} \right| = \sqrt{{(1 + 1)}^{2} + {( - 6 + 3)}^{2} + {(10 - 4)}^{2}} = 7$
$\left| \overrightarrow{\mathbf{CD}} \right| = \sqrt{{( - 1 - 5)}^{2} + {( - 3 + 1)}^{2} + {(4 - 5)}^{2}} = \sqrt{41}$
and $\left| \overrightarrow{\mathbf{DA}} \right| = \sqrt{{(5 - 7)}^{2} + {( - 1 + 4)}^{2} + {(5 - 7)}^{2}} = \sqrt{17}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.