📦 Volume of a Parallelepiped Calculator
Calculate volume from three vectors
Three Vectors (edges from one vertex)
Vector 1 (x, y, z)
Vector 2 (x, y, z)
Vector 3 (x, y, z)
How to Use This Calculator
Enter Three Vectors
Input the x, y, z components of three vectors representing edges from one vertex of the parallelepiped.
Calculate
Press "Calculate Volume" to find the volume using the scalar triple product |v₁ · (v₂ × v₃)|.
View Result
See the volume displayed in cubic units, along with the scalar triple product calculation.
Formula
Volume = |v₁ · (v₂ × v₃)|
Scalar triple product (absolute value)
Steps:
- Calculate cross product: v₂ × v₃
- Calculate dot product: v₁ · (v₂ × v₃)
- Take absolute value: Volume = |v₁ · (v₂ × v₃)|
Example: v₁ = (1, 0, 0), v₂ = (0, 1, 0), v₃ = (0, 0, 1)
v₂ × v₃ = (0, 0, 1)
v₁ · (v₂ × v₃) = 1×0 + 0×0 + 0×1 = 0
Wait, that's wrong! Let me recalculate...
v₂ × v₃ = (1×1 - 0×0, 0×0 - 0×1, 0×0 - 1×0) = (1, 0, 0)
v₁ · (v₂ × v₃) = 1×1 + 0×0 + 0×0 = 1
Volume = |1| = 1 cubic unit
About Volume of a Parallelepiped Calculator
The Volume of a Parallelepiped Calculator finds the volume of a parallelepiped (3D parallelogram) from three vectors using the scalar triple product. Volume = |v₁ · (v₂ × v₃)|.
When to Use This Calculator
- Geometry: Calculate volumes of parallelepipeds
- Physics: Find volumes in 3D space
- Engineering: Calculate volumes of 3D structures
- Mathematics: Apply scalar triple product
Understanding Parallelepiped Volume
A parallelepiped is a 3D shape with six parallelogram faces. Its volume equals the absolute value of the scalar triple product of three edge vectors meeting at one vertex.
Frequently Asked Questions
What is a parallelepiped?
A parallelepiped is a 3D shape with six parallelogram faces. It's like a "slanted box" where opposite faces are parallel parallelograms. Examples: rectangular box, rhomboid.
How is volume calculated?
Volume = |v₁ · (v₂ × v₃)|, where v₁, v₂, v₃ are three edge vectors from one vertex. The scalar triple product gives the volume (absolute value).
What if the volume is zero?
Zero volume means the three vectors are coplanar (lie in the same plane). The parallelepiped is "flat" (degenerate), so volume = 0.
Can I use any three vectors?
Yes! As long as they represent three edges from one vertex. The volume is the same regardless of which vertex you choose, as long as you use edges from that vertex.