Question

If the coordinate of points $P(2,3,4),Q(1,-2,1)$, and $O$ is the origin then:

• A. $OP=OQ$
• B. $OP\perp OQ$
• C. $OP\parallel OQ$
• D. None of these

Answer 1

The correct option is B

$OP\perp OQ$

Explanation for the correct option.

Step 1: Find the direction ratios

The direction cosines of the coordinate $\left(a,b,c\right)$ is $\left(\frac{a}{\sqrt{a^2+b^2+c^2}},\frac{b}{\sqrt{a^2+b^2+c^2}},\frac{c}{\sqrt{a^2+b^2+c^2}}\right)$

The direction ratio of line segment OP are given by

$\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}}$
So,${\mathrm{l}}_1=\frac{2}{\sqrt{29}},{\mathrm{m}}_1=\frac{3}{\sqrt{29}},{\mathrm{n}}_1=\frac{4}{\sqrt{29}}$

Step 2: Find the direction ratios

Direction ratios of the line segment $OQ$ are given by:
$\frac{1}{\sqrt{6}},\frac{-2}{\sqrt{6}},\frac{1}{\sqrt{6}}$
So, ${\mathrm{I}}_2=\frac{1}{\sqrt{6}},{\mathrm{m}}_2=\frac{-2}{\sqrt{6}},{\mathrm{n}}_2=\frac{1}{\sqrt{6}}$

Step 3: Determine perpendicularity

We know for perpendicular lines, $l_1{l}_2+m_1{m}_2+n_1{n}_2=0$.
$\begin{array}{rcl}{l}_1{l}_2+m_1{m}_2+n_1{n}_2& =& \frac{2}{\sqrt{29}}\times \frac{1}{\sqrt{6}}+\frac{3}{\sqrt{29}}\times \left(\frac{-2}{\sqrt{6}}\right)+\frac{4}{\sqrt{29}}\times \frac{1}{\sqrt{6}}\\ & =& \frac{2}{\sqrt{174}}-\frac{6}{\sqrt{174}}+\frac{4}{\sqrt{174}}\\ & =& 0\end{array}$

Hence, option B is correct.