The sign of [a b c] has geometric meaning:

• [a b c] > 0: the vectors a, b, c (in order) form a right-handed system. Curling from a to b and following the right-hand rule points along c (approximately).
• [a b c] < 0: left-handed system.
• [a b c] = 0: degenerate (coplanar).

For volume calculations, always take the absolute value: V = |[a b c]|.

For orientation problems (rare in JEE Main, occasional in Advanced): the sign determines whether a given point is "above" or "below" the plane of the other three.

If A, B, C define a plane, and D is a point, then [AB AC AD] > 0 means D is on one side of the plane, and [AB AC AD] < 0 means the other side. The sign depends on the ordering of A, B, C.