Triple Product
Scalar triple product gives parallelepiped volume.
Key idea
Scalar triple product gives parallelepiped volume.
In depth
The scalar triple product a · (b × c) is the signed volume of the parallelepiped spanned by the three vectors; it equals the determinant of the matrix of the vectors. Zero means the vectors are coplanar.
Worked example
For i, j, k: i · (j × k) = 1, a unit cube.
Formula
Real-world uses
- Computing the volume of a parallelepiped-shaped box or the volume of a crystal unit cell
Common mistake
Taking the absolute value when you need orientation, or the reverse.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Triple Product?
Scalar triple product gives parallelepiped volume. It is a topic at the expert level in Linear Algebra.
What is the formula for Triple Product?
The key formula is u·(v×w).
Where is Triple Product used in real life?
Real-world examples: Computing the volume of a parallelepiped-shaped box or the volume of a crystal unit cell.
What should I learn before Triple Product?
Make sure you are comfortable with Cross Product, the previous topic in Linear Algebra.
What is a common mistake with Triple Product?
Taking the absolute value when you need orientation, or the reverse.
How can I practise Triple Product?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Triple Product?
Move on to Lines and Planes in Vector Form next.
Related expert topics in Linear Algebra
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.
- Adjugate Matrix – Transpose of the cofactor matrix.