Question

Lines $OA,OB$ are drawn from $O$ with direction cosines proportional to $(1,-2,-1),(3,-2,3).$ Find the direction cosines of the normal to the plane $AOB$

• A. $⟨\pm \frac{4}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$
• B. $⟨\pm \frac{2}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$
• C. $⟨\pm \frac{8}{\sqrt{29}}\pm \frac{6}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$
• D. $⟨\pm \frac{8}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$

Answer 1

The correct option is A $⟨\pm \frac{4}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$

Let $ax+by+cz+d=0$ be the plane, then $O(0,0,0);A(1,-2,-1);B(3,-2,3)$
$\Rightarrow d=0$ and $a-2b-c=0$ also $3a-2b+3c=0$
Putting $c=(a-2b),$ we get $6a=8b$
$a=\frac{4}{3}b\therefore a=\frac{4}{3}b,b=b$ and $c=-\frac{2b}{3}$
And the plane is $b(\frac{4}{3}x+y-\frac{2}{3}z)=0$
$4x+3y-2z=0$
The normal $\pm (\overrightarrow{n}=4\hat{i}+3\hat{j}-2\hat{k})$
$\hat{n}=\pm (\frac{4}{\sqrt{29}}\hat{i}+\frac{3}{\sqrt{29}}\hat{j}+\frac{2}{\sqrt{29}}\hat{k})$
D.C' s of normal vector $⟨\pm \frac{4}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$