Question

Consider a pyramid $OPQRS$ located in the first octant $( x \geq 0, y \geq 0, z \geq 0)$ 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 $S T=3$
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 $(3 j )$ in $\perp$ direction $\frac{3}{2} \hat{ i }-\frac{3}{2} \hat{ j }+3 \hat{ k }$
$p =\frac{\left|3 \hat{ j } \times\left(\frac{3}{2} \hat{ i }-\frac{3}{2} \hat{ j }+3 \hat{ k }\right)\right|}{\left|\frac{3}{2} \hat{ i }-\frac{3}{2} \hat{ j }+3 \hat{ k }\right|}=\frac{\left|-\frac{9}{2} \hat{ k }+9 \hat{i}\right|}{\left|\frac{3}{2} \hat{ i }-\frac{3}{2} \hat{ j }+3 \hat{ k }\right|}$
$\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}} $
$=\frac{\sqrt{45}}{\sqrt{6}}=\frac{\sqrt{15}}{\sqrt{2}}$