Question

The $\perp$ distance of a corner of a unit cube on a diagonal not passing through is

• A. $\frac{\sqrt{6}}{3}$
• B. $3\sqrt{3}$
• C. $2\sqrt{3}$
• D. None of these

Answer 1

The correct option is A $\frac{\sqrt{6}}{3}$

Let the edges Of unit cube are OA along z axis,OB along x axis and OC along y axis.

Let CM be the perpendicular from corner C on diagonal OP where point P is (1,1,1)

OP = $\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$

Unit vector in the direction of OP is = $\frac{\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}}{\sqrt{3}}$

OM= projection of OC on OP

OM= OC . (unit vector of OP)

OM=$\overrightarrow{j}\frac{\left(\right(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k})}{\sqrt{3}}$=$1\sqrt{3}$

Now $C{M}^2=O{C}^2-O{M}^2$

$C{M}^2$=$1-\frac{1}{3}$ = $\frac{2}{3}$

So CM =$\frac{\sqrt{2}}{\sqrt{3}}$ = $\frac{\sqrt{6}}{3}$