Volume of a Parallelepiped Calculator

Calculate the volume using the scalar triple product of three edge vectors.

Enter Edge Vectors

Vector a
Vector b
Vector c

Result

Volume
--
cubic units
Cross Product (b x c)--
Dot Product a . (b x c)--
|Scalar Triple Product|--

Step-by-Step Solution

Understanding the Parallelepiped

A parallelepiped is a three-dimensional figure formed by six parallelogram faces. It is the 3D analog of a parallelogram. Its volume can be computed using the scalar triple product of its three defining edge vectors.

The Scalar Triple Product

Given three vectors a, b, and c, the scalar triple product is defined as a . (b x c). The absolute value of this product gives the volume of the parallelepiped spanned by the three vectors.

Cross Product (b x c)

Produces a vector perpendicular to both b and c.

(b_y*c_z - b_z*c_y, b_z*c_x - b_x*c_z, b_x*c_y - b_y*c_x)

Dot Product

Scalar product of vector a with the cross product result.

a_x*d_x + a_y*d_y + a_z*d_z

Volume Formula

Absolute value of the scalar triple product.

V = |a . (b x c)|

Properties of Parallelepipeds

  • A parallelepiped has 6 faces, 8 vertices, and 12 edges.
  • Opposite faces are congruent parallelograms.
  • A rectangular parallelepiped (cuboid) has all right angles.
  • If the scalar triple product is zero, the three vectors are coplanar and form no volume.

Applications

Parallelepiped volume calculations are used in physics (determining volumes in crystal lattice structures), computer graphics (bounding volumes), and linear algebra (testing linear independence of vectors).