Question

I. lf $P=(O,1,2),Q=(4,-2,1)$, then $\angle POQ=\pi /2$ where $O$' is origin.
II. lf the $d.r$'s of two lines are $(1,-1,0)$ and $(1,-2,1)$, then the angle between them is $\frac{\pi }{6}$.
Which of fhe above statements are correct

• A. only I
• B. only II
• C. Both I & II
• D. Neither I nor II.

Answer 1

The correct option is C Both I & II

I) Taking the product of the Drs
$\left(0\right)\left(4\right)+\left(1\right)(-2)+\left(2\right)\left(1\right)=0$
Thus the lines are perpendicular.
Hence, I is correct.
II) Taking dot product of the Drs, we get
$\left(1\right)\left(1\right)+(-2)(-1)+\left(1\right)\left(0\right)=\left(\sqrt{2}\right)\left(\sqrt{6}\right)\cos \theta$
$\Rightarrow 3=\sqrt{12}\cos \theta$
$\Rightarrow 3=2\sqrt{3}\cos \theta$
$\Rightarrow \cos \theta =\frac{\sqrt{3}}{2}$
$\Rightarrow \theta =\frac{\pi }{6}$