Question

If the projection of any line on co-ordinate axes are $3,4,$ and $5$ then its length is

• A. $12$
• B. $50$
• C. $5\sqrt{2}$
• D. $3\sqrt{2}$

Answer 1

The correct option is C

$5\sqrt{2}$

Explanation of the correct option.

Computing the length of the line.

Let $d$ be the length of the line, then

The projection on $x$-axis $=dl=3$ $.....\left(1\right)$

The projection on $y$-axis $=dm=4$ $.....\left(2\right)$

The projection on $z$-axis $=dn=5$ $.....\left(3\right)$

Squaring $\left(1\right)$, $\left(2\right)$, and $\left(3\right)$ and adding, we get

$d^2\left(l^2+m^2+n^2\right)=3^2+4^2+5^2$

$\Rightarrow$ $d^2\left(l^2+m^2+n^2\right)=9+16+25$

$\Rightarrow$ $d^2\left(1\right)=50$ $\left[\because l^2+m^2+n^2=1\right]$

$\Rightarrow$ $d=\sqrt{50}$

$\Rightarrow$ $d=5\sqrt{2}$

Hence option $\left(C\right)$ is the correct option.