Volume of a Parallelepiped Calculator – Guide & Formulas
Calculate the volume of a parallelepiped defined by three vectors using the scalar triple product. Get step-by-step solutions.
Calculate the volume of a parallelepiped (3D parallelogram-shaped solid) defined by three edge vectors using the scalar triple product.
Key Takeaway
Use the free Volume of a Parallelepiped Calculator to calculate the volume of a parallelepiped defined by three vectors using the scalar triple product. get step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Volume of a Parallelepiped Calculator
- Enter the components of the first edge vector a⃗.
- Enter the components of the second edge vector b⃗.
- Enter the components of the third edge vector c⃗.
- Click "Calculate" to find the volume using the scalar triple product.
The Formula
Variable Definitions
- V: Volume of the parallelepiped
- a⃗, b⃗, c⃗: Three edge vectors defining the parallelepiped
- ×: Cross product operation
- ·: Dot product operation
- |...|: Absolute value (volume is always positive)
Example: Volume from Vectors a=(1,0,0), b=(0,1,0), c=(0,0,1)
Find the volume of a parallelepiped with edge vectors a = (1,0,0), b = (0,1,0), c = (0,0,1).
- Step 1: Calculate b × c = (0,1,0) × (0,0,1) = (1×1 - 0×0, 0×0 - 0×1, 0×0 - 1×0) = (1, 0, 0).
- Step 2: Calculate a · (b × c) = (1,0,0) · (1,0,0) = 1×1 + 0×0 + 0×0 = 1.
- Step 3: Volume = |1| = 1 cubic unit.
- Step 4: This is a unit cube with side length 1.
- Step 5: Verification: The determinant of the matrix [a;b;c] = det([[1,0,0],[0,1,0],[0,0,1]]) = 1.
Frequently Asked Questions
What is a parallelepiped?
A parallelepiped is a 3D figure formed by six parallelogram faces. It is the 3D analog of a parallelogram, with three pairs of parallel faces.
What is the scalar triple product?
The scalar triple product a⃗ · (b⃗ × c⃗) gives the signed volume of the parallelepiped formed by vectors a, b, and c. Its absolute value is the volume.
How is the cross product involved?
First, b⃗ × c⃗ gives a vector perpendicular to both b and c with magnitude equal to the area of the parallelogram they form. The dot product with a then gives the volume.
Can the volume be negative?
The scalar triple product can be negative if the vectors form a left-handed coordinate system. Volume is always taken as the absolute value.
What if the volume is zero?
A volume of zero means the three vectors are coplanar (lie in the same plane) or one is a multiple of another. The parallelepiped is degenerate (flat).
How does this relate to the determinant?
The scalar triple product equals the determinant of the 3×3 matrix formed by the three vectors as rows (or columns): det([a;b;c]).
What is the area of a face?
The area of the face formed by two vectors is given by their cross product magnitude: |b⃗ × c⃗| gives the area of the parallelogram formed by b and c.
Can I use this for a rectangular box?
Yes, a rectangular box (cuboid) is a special case of a parallelepiped where the edge vectors are perpendicular. The volume is simply length × width × height.
What are real-world applications?
Parallelepiped volume is used in crystallography (unit cell volume), physics (flux calculations), and computational geometry.
How do I find the volume from coordinates?
If you have three edge vectors from a common vertex, enter their components into this calculator. The scalar triple product will give the volume.