Calculation of Areas and Volumes
Area of Parallelogram
Geometrically, the magnitude of the cross product of two vectors coincides with the area of the parallelogram whose sides are those vectors.
Example
Given the vectors 
and 
, find the area of the parallelogram whose sides are the vectors 
and 
·
Area of a Triangle
Example
Determine the area of the triangle whose vertices are the points A = (1, 1, 3), B = (2, −1, 5) and C = (−3, 3, 1).
Volume of a Parallelepiped
Geometrically, the absolute value of the triple product represents the volume of the parallelepiped whose edges are the three vectors that meet in the same vertex.
Find the volume of the parallelepiped formed by the following vectors:
Volume of a Tetrahedron
The volume of a tetrahedron is equal to 1/6 of the absolute value of the triple product.
Calculate the volume of the tetrahedron whose vertices are the points A = (3, 2, 1), B = (1, 2, 4), C = (4, 0, 3) and D = (1, 1, 7).
