Thank you for reporting, we will resolve it shortly
Q.
The perpendicular distance of a corner of a unit cube form a diagonal not passing through it is
Vector Algebra
Solution:
Let the unit vectors $i, j$ and $k$ be denoted by the coterminous edges $O A, O B$ and $O C$, respectively of the unit cube.
Let $CN$ be the perpendicular drawn from $C$ on the diagonal $O E$ of the cube which does not pass through $C$.
Here, $O E=i+j+$ $k$.
Let $e$ be the unit vector along $O E$. Then,
$ e =\frac{i+j+k}{|i+j+k|}=\frac{i+j+k}{\sqrt{(1+1+1)}} $
$=\frac{1}{\sqrt{3}}(i+j+k) $
$\therefore O N=$ projection of $O C$ on $O E$
$=k \cdot e=k \cdot \frac{1}{\sqrt{3}}(i+j+k)=\frac{1}{\sqrt{3}}$
$\therefore C N^{2}=O C ^{2}-O N^{2}$
[In right triangle $\Delta O C N$ ]
$=1^{2}-\left(\frac{1}{\sqrt{3}}\right)^{2}=1-\frac{1}{3}=\frac{2}{3}$
$\therefore C N=\sqrt{\frac{2}{3}}=\frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{6}}{3}$
