The determinant of a 3×3 matrix is signed volume. This chapter shows how to read its sign by choosing one column as the viewing vector, projecting the other two columns onto the orthogonal plane and comparing the visible counterclockwise order with the cyclic column order. The same idea then leads to a recursive projection rule and to a visual explanation of cofactor signs.
Determinant sign is orientation, not just a plus or minus attached to volume.
The later pages extend this orientation rule beyond 3D and connect it to the alternating signs in cofactor expansion.
Projecting onto the plane perpendicular to one column removes the viewing direction but keeps the orientation information needed to decide sign. From the tip of the viewing vector, the remaining two projected arrows either match the cyclic column order or reverse it.
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In expansion along column 1, term i starts with the pivot row direction followed by the row directions of the minor. Restoring the standard order requires i−1 swaps, so even numbers of swaps keep the sign and odd numbers of swaps negate it.