Question

Consider a pyramid $OPQRS$ located in the first octant $(x\ge 0,y\ge 0,z\ge 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis respectively. The base $OPQR$ of the pyramid is a square with $OP=3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS=3$. Then

• A. the acute angle between $OQ$ and $OS$ is $\frac{\pi }{3}$
• B. the equation of the plane containing the triangle $OQS$ is $x-y=0$
• C. the length of the perpendicular from $P$ to the plane containing the triangle $OQS$ is $\frac{3}{\sqrt{2}}$
• D. the perpendicular distance from $O$ to the straight line containing $RS$ is $\sqrt{\frac{15}{2}}$

Answer 1

Answer: (B), (C), (D)

acute angle between $OQ$ \& OS
(A) Distance between $OT=\frac{3}{\sqrt{2}}$
Distance between $ST=3$
Clearly angle between OT and OS
$\tan \theta =\frac{3}{3/\sqrt{2}}=\sqrt{2}$
$\tan \theta =\sqrt{2}$
$\theta ={\tan }^{-1}(\sqrt{2})$
Option A is wrong
(B) Clearly equation of plane $OQS$ is $x-y=0$
(for each point $x=y$ )
(C) Equation of plane $OQS$ is $x-y=0$
Distance from $P(3,0,0)=\frac{13-01}{\sqrt{1+1}}=\frac{3}{\sqrt{2}}$
(D)
$p=$ projection of $(3j)$ in $\perp$ direction $\frac{3}{2}\hat{i}-\frac{3}{2}\hat{j}+3\hat{k}$
$p=\frac{|3\hat{j}\times \left(\frac{3}{2}\hat{i}-\frac{3}{2}\hat{j}+3\hat{k}\right)|}{|\frac{3}{2}\hat{i}-\frac{3}{2}\hat{j}+3\hat{k}|}=\frac{|-\frac{9}{2}\hat{k}+9\hat{i}|}{|\frac{3}{2}\hat{i}-\frac{3}{2}\hat{j}+3\hat{k}|}$
$\Rightarrow \frac{\sqrt{\frac{81}{4}+81}}{\sqrt{\frac{9}{4}+\frac{9}{4}+9}}=\frac{\sqrt{\frac{9}{4}+9}}{\sqrt{\frac{1}{4}+\frac{1}{4}+1}}$
$=\frac{\sqrt{45}}{\sqrt{6}}=\frac{\sqrt{15}}{\sqrt{2}}$