How can I verify that points form a tilted box? Given the points $$ P = (1, 0, -1) \\\ Q = (1, 1, 1) \\\ R = (2, 2, 1) \\\ S = (2, 1, -1) \\\ $$ Choose $T, U, V$ such that $OPQRSTUV$ is a tilted box. A possible answer is apparently $$ O = (0, 0, 0) \\\ T = (0, 0, 2) \\\ U = (1, 2, 2) \\\ V = (1, 1, 0) \\\ $$ This plot shows the points $PQRS$ in blue, $O$ in red and $TUV$ in black , the points really don't look like a tilted box. How can I verify that the answer really is a tilted box?

Assuming "box" means "right rectangular prism," the answer supplied is incorrect.

Notice that $OP$ and $OQ$ are perpendicular, since $$(1, 0, -1)\cdot(1, 1, 1) = 1 + 0 - 1 = 0.$$

So either $P, O, Q$ are three consecutive vertices of one of the rectangular faces of the box, or they are three consecutive vertices of one of the rectangles consisting of two opposite edges of the box. In either case, the fourth vertex of this rectangle, which is $$(1, 0, -1) + (1, 1, 1) = (2, 1, 0)$$ is a vertex of the box. Since $(2, 1, 0)$ doesn't appear, these points certainly aren't the vertices of a right rectangular prism.

**EDIT** : I used a computer to check, and there are no other right angles among the five points given. That means that $O$, $P$, $Q$, $R$, and $S$ are not five of the eight vertices of _any_ right rectangular prism.