IIT-JEE 2004Question 1
Find the equation of the plane passing through $(1,1,1)$ and parallel to the lines $L_1,L_2$ having direction ratios $(1,0,-1)$, $(1,-1,0)$. Find the volume of the tetrahedron formed by the origin and the points where this plane intersects the coordinate axes.
3D Geometry
AnswerPlane: $x+y+z=3$; volume $=\dfrac92$ cubic units
Solution
Normal $=(1,0,-1)\times(1,-1,0)=(-1,-1,-1)$, so the plane through $(1,1,1)$ is $x+y+z=3$ with intercepts $3,3,3$. Volume $=\frac16\cdot3\cdot3\cdot3=\frac92$.