Question

The co-ordinates of the point $P$ are $(x,y,z)$ and the direction cosines of the line $OP$ are $l,m,n$ when $O$ is the origin, . If$OP=r$, then

• A. $l=x,m=y,n=z$
• B. $l=xr,m=yr,n=zr$
• C. $x=lr,y=mr,z=nr$
• D. None of these

Answer 1

The correct option is C

$x=lr,y=mr,z=nr$

Explanation for correct option:

Finding the values:

Given that $P\to \left(x,y,z\right)$ and $O\to \left(0,0,0\right)$

New direction cosines are $l,m,n$.

We know that, direction cosines are written as

$l=\frac{x}{\sqrt{x^2+y^2+z^2}}\to \left(i\right)$

$m=\frac{y}{\sqrt{x^2+y^2+z^2}}\to \left(ii\right)$

$n=\frac{z}{\sqrt{x^2+y^2+z^2}}\to \left(iii\right)$

Also,$OP=r$

We know that distance between $O$ and $P$ is calculated as

$$\begin{gathered} r=\sqrt{{\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-0\right)}^2} \\ =\sqrt{x^2+y^2+z^2} \end{gathered}$$

Substituting the value of $r$ in equations $\left(i\right)$,$\left(ii\right)$ and $\left(iii\right)$

Therefore

$l=\frac{x}{r}$$\Rightarrow x=lr$

$m=\frac{y}{r}$$\Rightarrow y=mr$

$n=\frac{z}{r}$$\Rightarrow z=nr$

Hence, option (C) is the correct answer