If the plane faces of a tetrahedon are represented by the equations $\vec{r} \cdot (\vec{i} + \vec{j}) = 0$, $\vec{r} \cdot (n\vec{k} + m\vec{j}) = 0$, $\vec{r} \cdot (m\vec{k} + \vec{i}) = 0$ and $\vec{r} \cdot (\vec{i} + m\vec{j} + n\vec{k}) = p$, then the volume of the tetrahedon is:
Step-by-Step Solution
Key Concept: Volume of tetrahedron formed by four planes is computed using scalar triple product of position vectors of vertices.
Given four planes $lx + my = 0$, $my + nz = 0$, $lx + nz = 0$, and $lx + my + nz = p$, solving the first three simultaneously gives the point $A\left(\frac{p}{l}, -\frac{p}{m}, \frac{p}{n}\right)$. Similarly, $B\left(\frac{p}{l}, \frac{p}{m}, -\frac{p}{n}\right)$, $C\left(-\frac{p}{l}, -\frac{p}{m}, \frac{p}{n}\right)$, and $O(0,0,0)$. Volume of tetrahedron is $\frac{1}{6}\left|\frac{4p^3}{lmn}\right| = \frac{2p^3}{3lmn}$.
Correct Answer: 4