The Scalar Triple Product and Volume
Simple Explanation
Combining the dot and cross products, the scalar triple product a·(b×c) gives (up to a sign) the volume of the parallelepiped (a "slanted box") formed by three vectors sharing a common vertex: Volume = |a·(b×c)|.
Why Do We Need It?
This is the natural 3D extension of using the cross product for area — extending from a flat 2D shape (a parallelogram) to a full 3D solid (a parallelepiped).
Formula
The Scalar Triple Product and Volume
Volume = |a · (b × c)|
The volume of the parallelepiped (a 3D "slanted box") formed by three vectors a, b, and c, sharing a common vertex.
- a, b, c
- — the three edge vectors of the parallelepiped, from one shared corner
When to use it: Whenever the volume of a parallelepiped formed by three vectors is needed.
Worked Example
Find a volume using the scalar triple product
Find the volume of the parallelepiped formed by a=(1,0,0), b=(0,1,0), and c=(0,0,1).
Why Does This Work?
|b×c| gives the area of the base parallelogram formed by b and c, and the direction of b×c is perpendicular to that base — so a·(b×c) essentially measures (base area) × (a's component perpendicular to the base), which is exactly base area × height, the formula for the volume of a parallelepiped.
Real-Life Example
Finding the volume of a slanted storage container
An engineer describes a storage tank shaped like a parallelepiped (a "slanted box") using three edge vectors from one corner.
The scalar triple product gives the tank's actual volume directly from these three edge vectors.
Practice
Find the volume of the parallelepiped formed by a=(2,0,0), b=(0,3,0), and c=(0,0,4).
HardCommon mistake
Forgetting to take the absolute value — the scalar triple product can come out negative depending on the vectors' orientation, but a volume must always be non-negative.
Quick Review
- Volume = |a·(b×c)|.
- Combines the cross product (base area) with the dot product (accounting for height).
- The 3D extension of using the cross product for a parallelogram's area.