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.

Cross Product of Two Vectors

Area of a Parallelogram

Example

Given the vectors Vector Area Example and Vector Area Example, find the area of the parallelogram whose sides are the vectors Vector U and Vector V·

Vectorial Product

Vector Area Solution

Area of a Triangle

Area of a Triangle

Area of a Triangle Formula

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).

Vectorial Product

Vectorial Area Operations

Vectorial Area Calculations

Vectorial Area Operations

Vectorial Area Solutions


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:

Vectorial Volume Example

Volume of a Parallelepiped

Volume of a Tetrahedron

The volume of a tetrahedron is equal to 1/6 of the absolute value of the triple product.

Volume of a Tetrahedron

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).

Vectorial Volume Example

Vectorial Volume Example

Vectorial Volume Example

Volume of a Tetrahedron

Mixed Product Operations

Vectorial Volume Solution