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Triple products
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Scalar triple product gives the volume of a parallelepiped.
A·(B×C)The scalar triple product A·(B×C) equals the volume of the parallelepiped spanned by the three vectors and is zero if they lie in one plane. The vector triple product obeys A×(B×C) = B(A·C) − C(A·B).
For î, ĵ, k̂ the triple product is 1: a unit cube.
Brackets matter: A×(B×C) is different from (A×B)×C.
In more detail
A scalar triple product A·(B×C) gives the volume of the parallelepiped formed by three vectors. A vector triple product A×(B×C) follows a special identity useful in electromagnetism and rotation.
Real-world uses
- The volume of a crystal unit cell is found with a scalar triple product of its edge vectors.
- The vector triple product appears when deriving magnetic fields and orbital motion.
- The volume of a parallelepiped-shaped crystal cell is given by the scalar triple product of its three edge vectors.
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Frequently asked questions
What does a zero triple product mean?
The three vectors lie in the same plane, so they do not span a volume.
Do I need the vector triple product for exams?
It appears in advanced courses, mainly in electromagnetism and rotational dynamics.
What does a zero triple product tell me?
It means the three vectors lie in one plane, so they do not enclose any volume.
Related topics in Measurement & Math Tools
Unit vectors · Dot product · Cross product · Calculus: derivatives · Calculus: integrals · Partial derivatives