Mathematics

Problem

Statement I Distance of point

D(1, 0, -1)

from the plane of points

A(1, -2, 0)

B(3, 1, 2)

and

C(-1, 1, -1)

\frac{8}{\sqrt{229}}

. Statement II Volume of tetrahedron formed by the points

A

B

C

and

D

\frac{\sqrt{229}}{2}

Intuitive explanation

The key is recognizing that both statements hinge on the same geometric object: the plane containing $A$ , $B$ , $C$ . Find the plane's equation using the normal vector $\vec{AB} \times \vec{AC}$ , then use the point-to-plane distance formula for Statement I. For Statement II, the tetrahedron volume equals $\frac{1}{6}|\vec{AD} \cdot (\vec{AB} \times \vec{AC})|$ —notice this shares the cross product, and the distance formula connects directly to volume through the base area. Check whether both statements are consistent with each other.

Intuitive explanation

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