Sarrus Rule Calculator guide
How Determinants Give You Volume and the Cross Product
If you place three 3D vectors as the rows of a 3×3 matrix, the absolute value of its determinant is the volume of the parallelepiped they span. A determinant of zero means the three vectors are coplanar, so the box is flat and has no volume. The sign tells you the orientation of the three vectors.
Start with the simplest possible case
The matrix loaded into the calculator has the vectors (2, 0, 0), (0, 3, 0) and (0, 0, 4) as its rows. Each one points straight along one of the three axes, so the shape they outline is an ordinary rectangular box measuring 2 by 3 by 4.
The volume of that box is 2 × 3 × 4 = 24. And the determinant is 24.
That is the whole idea. What makes it useful is that it keeps working when the vectors are not at right angles — the box leans over into a slanted shape called a parallelepiped, but the determinant still gives you its volume.
The scalar triple product is the same thing
You may have met this written as a · (b × c) — take the cross product of two of your vectors, then dot the result with the third. It has a name: the scalar triple product.
It is worth being clear that this is not merely related to the determinant. It is the determinant of the matrix whose rows are a, b and c — the same number, computed by a different route.
So if a question gives you three vectors and asks for a volume, you can either grind through a cross product and then a dot product, or you can stack the vectors into a matrix and take one determinant. The second is usually less work.
What the sign is telling you
A positive determinant means your three vectors form a right-handed set — the same handedness as the standard x, y, z axes. A negative determinant means they are left-handed, a mirror image of that.
This is why volume is the absolute value of the determinant rather than the determinant itself. A box cannot have negative volume. The minus sign is not describing size at all; it is describing orientation, and it is genuine information as long as you know to read it that way.
Zero volume means the vectors are coplanar
If the determinant comes out as zero, all three vectors lie in a single flat plane. The box they were supposed to outline has been squashed to zero thickness, so there is no volume left.
This gives you a quick test that is worth remembering: to check whether three vectors lie in the same plane, stack them into a matrix and take the determinant. Zero means coplanar.
It is also the same fact as linear dependence from the other guides, just seen as a picture. Dependent rows and coplanar vectors are two descriptions of one situation.
