I'm studying linear algebra and, in particular, the concept of determinant. I've understood the determinant in 2 dimension, but in 3d i have not clear the concept of oriented volume. Can you explain me ?
2026-04-11 23:00:09.1775948409
Oriented Volume and determinant
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In 3d the absolute value of the determinant corresponds to the volume of the parallelepiped formed by the rows constructed from the vectors $r_1$, $r_2$, and $r_3$.
Since the determinant is equal to the triple product $$r_1\cdot (r_2\times r_3)$$ without absolute value the determinant can be negative or positive depending upon the orientation of the row vectors.