Question

The perpendicular distance of a corner of a unit cube form a diagonal not passing through it is

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

Answer 1

Answer: (B)

Let the unit vectors $i,j$ and $k$ be denoted by the coterminous edges $OA,OB$ and $OC$, respectively of the unit cube.
Let $CN$ be the perpendicular drawn from $C$ on the diagonal $OE$ of the cube which does not pass through $C$.
Here, $OE=i+j+$ $k$.
Let $e$ be the unit vector along $OE$. Then,

$e=\frac{i+j+k}{|i+j+k|}=\frac{i+j+k}{\sqrt{(1+1+1)}}$
$=\frac{1}{\sqrt{3}}(i+j+k)$
$\therefore ON=$ projection of $OC$ on $OE$
$=k\cdot e=k\cdot \frac{1}{\sqrt{3}}(i+j+k)=\frac{1}{\sqrt{3}}$
$\therefore C{N}^2=O{C}^2-O{N}^2$
[In right triangle $\Delta OCN$ ]
$=1^2-{\left(\frac{1}{\sqrt{3}}\right)}^2=1-\frac{1}{3}=\frac{2}{3}$
$\therefore CN=\sqrt{\frac{2}{3}}=\frac{\sqrt{2}}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{6}}{3}$