The scalar triple product turns three three-dimensional vectors into one signed volume. Its absolute value is their parallelepiped’s volume.
Use it to test coplanarity and calculate oriented three-dimensional volume.
The relationship
The defining relationship
u·(v×w)
See the structure
What the calculation is doing
1234
row 1 · column 1
2012
1×2 + 2×1 = 4
Worked interpretation
Read the result in context
u·(v×w)
Interpret with care
Important boundary
Changing the order can change the sign; a zero result means the vectors are coplanar or dependent.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
Enter three three-dimensional vectors.
Preserve the requested vector order.
Read the signed scalar and its non-negative volume magnitude.
Calculation method
Cross two vectors, then take a dot product
The scalar triple product a · (b × c) is the signed volume of the parallelepiped formed by the ordered vectors. Its absolute value is the geometric volume.
Worked example
Standard basis volume
Use the three standard basis vectors in cyclic order.
⟨1,0,0⟩ · (⟨0,1,0⟩ × ⟨0,0,1⟩) = 1
Supported inputs
Precision and limits
Vector size
General vectors may contain 2–8 components; three-dimensional products require exactly three.
Entry precision
Each signed decimal may contain up to 30 digits and 15 decimal places.
Square roots
Squared magnitudes remain exact. Irrational roots and normalized components are explicitly labelled approximations.
