Triple Products and Volume

Calculus 3 · Axiom Academy

LESSON Triple Products and Volume One number, the scalar triple product , measures the volume of the box that three vectors span. Send three vectors , , and out from one corner. Slide copies of each along the others and they close up into a parallelepiped — a box whose faces are parallelograms. The scalar triple product bundles these three vectors into one number by crossing two and dotting with the third. Cross two vectors, then dot with the third — the result is a single scalar Equivalently, the determinant of the rows , , Why should a determinant measure volume? Take : it points straight out of the base and its length equals the base's area . Dotting with multiplies that area by the part of standing perpendicular to the base — the height . Area times height is exactly the volume, so the absolute value of the scalar triple product is the box's volume. The length is the area of the parallelogram base spanned by and . Dotting with the unit normal picks off how far the box rises above its base. The raw value can be negative — its sign records the orientation of , , . Taking the absolute value discards the sign and leaves the true, positive volume. For the axis-aligned edges , , the determinant is , matching the rectangular block. The unit cube gives volume 1 .

This is the written version of the interactive lesson above. See the full Calculus 3 course.