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Mathematics

The Scalar Triple Product and Volume of Parallelepipeds

Quick fact

The volume of a parallelepiped spanned by three vectors is given by the absolute value of their scalar triple product: |a · (b × c)|. If the triple product is zero, the three vectors are coplanar, meaning the parallelepiped collapses to zero volume.

Why this is interesting

What if a single number could tell you the volume of a skewed box formed by any three arrows in space? The scalar triple product does exactly that—and it even reveals which way the box is oriented.

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