Question

A plane is parallel to two lines whose direction ratios are $(1,0,-1)$ and $(-1,1,0)$ and it passes through the point $(1,1,1)$, cuts the axis at $A,B,C$, then find the volume of the tetrahedron $OABC$.

Answer 1

Answer: 4.50

Let equation of plane
$a(x-1)+b(y-1)+c(z-1)=0$
and $a-c=0$ ...(1)
$-a+b=0$
$a=b=c=\lambda ,$ from $(1)$
$\lambda (x-1)+\lambda (y-1)+\lambda (z-1)=0$
$x+y+z=3$

Volume $=\frac{1}{6}[\overrightarrow{OA}\overrightarrow{OB}\overrightarrow{OC}]$
$=\frac{1}{6}\left|\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right|$
$=\frac{3\times 3\times 3}{6}=\frac{9}{2}$