Scalar triple products

Scalar triple products: linear algebra practice

Use scalar triple products to combine a cross-product area vector with a perpendicular height. Questions develop signed volume and orientation, calculate ordinary parallelepiped and tetrahedron volumes, and test whether three vectors lie in a plane through the origin. Coordinate examples include offset vertices and degenerate cases, with no determinant notation required.

What you will practise

• Compute a scalar triple product by a cross product followed by a dot product
• Interpret signed volume, ordinary volume and coplanarity
• Track cyclic order, swaps, scaling and tetrahedron volume

How to use this questionset

Prerequisites: Cross products, dot products, vector norms, edge differences and area from a cross product. The questions progress from direct interpretation to application and reasoning. Work out an answer before comparing the choices; for multi-select questions, evaluate every statement independently. Worked explanations show the reasoning and address relevant misconceptions, so an incorrect attempt can guide your next review.

Ordinary volume is the absolute value of the signed result. Coplanarity here concerns the three displacement vectors through a common origin; for four points, subtract one anchor to form the three edges. This geometric interpretation prepares the later study of determinants as oriented volume scale factors. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.