Parallelepiped Volume Calculator – Scalar Triple Product
Calculate parallelepiped volume from three edge vectors
Table of Contents
How to Use
- Enter the x, y, and z components of vector a (first edge)
- Enter the x, y, and z components of vector b (second edge)
- Enter the x, y, and z components of vector c (third edge)
- Click calculate to see the volume result
What is a Parallelepiped?
A parallelepiped is a three-dimensional figure formed by six parallelograms. It's the 3D analog of a parallelogram. When defined by three vectors a, b, and c emanating from a common vertex, the volume equals the absolute value of the scalar triple product.
The volume formula is: V = |a · (b × c)|, where b × c is the cross product and a · (b × c) is the dot product of a with that result.
The Scalar Triple Product
The scalar triple product a · (b × c) can also be computed as the determinant of a 3×3 matrix with vectors a, b, c as rows (or columns).
- Geometric meaning: The signed volume of the parallelepiped
- Sign: Positive if a, b, c form a right-handed system; negative if left-handed
- Zero result: Indicates the three vectors are coplanar (lie in the same plane)
Applications
- Crystallography: Computing unit cell volumes in crystal structures
- Physics: Calculating flux and field quantities
- Engineering: Volume calculations in structural analysis
- Computer Graphics: 3D collision detection and spatial calculations
- Linear Algebra: Testing linear independence of vectors
Special Cases
- Rectangular box: When vectors are mutually perpendicular, volume = |a| × |b| × |c|
- Cube: When all three vectors have equal length and are perpendicular
- Degenerate case: Volume = 0 when vectors are coplanar
Frequently Asked Questions
- What is the difference between a parallelepiped and a rectangular box?
- A rectangular box (cuboid) is a special case of a parallelepiped where all angles are 90 degrees. A general parallelepiped can have faces that are non-rectangular parallelograms with oblique angles.
- Why can the scalar triple product be negative?
- The sign indicates the orientation of the three vectors. A positive value means they form a right-handed coordinate system, while negative means left-handed. For volume, we take the absolute value since volume is always positive.
- What does it mean if the volume is zero?
- A zero volume indicates that the three vectors are coplanar—they all lie in the same plane. This means they don't span a 3D space and cannot form a proper parallelepiped.
- How is this related to the determinant?
- The scalar triple product equals the determinant of the 3×3 matrix formed by the three vectors. This connection is fundamental in linear algebra and explains why the determinant measures 'volume scaling' of linear transformations.