Question

If $O$ is the origin and $OP=3$ with direction ratios $-1,2,-2$, then co-ordinates of $P$ are

• A. $\left(1,2,2\right)$
• B. $\left(-1,2,-2\right)$
• C. $\left(-3,6,9\right)$
• D. $\left(\frac{-1}{3},\frac{2}{3},\frac{-2}{3}\right)$

Answer 1

The correct option is B

$\left(-1,2,-2\right)$

Explanation for the correct option:

Step 1 : given information

Here, the distance between the origin and point $P$ will be denoted by $d$ which is equal to $3$.

Now, suppose the coordinates of the point $P$ are $\left(ld,md,nd\right)$, where $l,m,n$ are the cosine ratios.

Denote the direction ratio $-1,2,-2$ as $a,b,c$. Then the cosine ratios are given by,

$\mathit{l}\mathbf{=}\frac{\mathbf{a}}{\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{+}{\mathbf{c}}^{\mathbf{2}}}}\mathbf{,}\mathbf{}\mathit{m}\mathbf{=}\frac{\mathbf{b}}{\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{+}{\mathbf{c}}^{\mathbf{2}}}}\mathbf{,}\mathbf{}\text{and}\mathit{n}\mathbf{=}\frac{\mathbf{c}}{\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{+}{\mathbf{c}}^{\mathbf{2}}}}$

Step 2 : finding the coordinates of the point $P$

The cosine ratios will be,

$\begin{array}{rcl}l& =& \frac{-1}{\sqrt{{\left(-1\right)}^2+{\left(2\right)}^2+{\left(-2\right)}^2}}\\ & =& \frac{-1}{\sqrt{1+4+4}}\\ & =& \frac{-1}{\sqrt{9}}\\ & =& \frac{-1}{3}\end{array}$

Similarly, $m\text{and}n$ will be $\frac{2}{3}\text{and}\frac{-2}{3}$

Since, the coordinates of the point $P$ are $\left(ld,md,nd\right)$, then

$\begin{array}{rcl}P& =& \left(\frac{-1}{3}\left(3\right),\frac{2}{3}\left(3\right),\frac{-2}{3}\left(3\right)\right)\\ & =& \left(-1,2,-2\right)\end{array}$

Hence, the correct option is (B).