Applications: area, volume and moment of force

The cross-product and the volume of a rectangular prism

The cross-product of three vectors u, v and w is the scalar defined by [u, v, w] = u . (v ^ w). Its absolute value gives the volume of the parallelepiped constructed from the three vectors:

Volume = |u · (v · w)|

For a tetrahedron ABCD, the volume is V = (1/6) * |AB · (AC · AD)|.

Example

Let u = (1,0,0), v = (0,1,0), w = (0,0,2). We calculate v ^ w = (12 - 00, 00 - 02, 00 - 10) = (2, 0, 0). Then u . (v^w) = 1*2 + 0 + 0 = 2. The volume of the parallelepiped is therefore 2 (consistent with a 1x1x2 block).

Collinearity, coplanarity

Three vectors u, v, w are coplanar if and only if their cross-product is zero: [u, v, w] = 0. This is a widely used criterion for checking whether four points lie in the same plane.

The moment of a force (physical application)

In mechanics, the moment of a force F applied at a point P relative to a point O is M = OP × F. This vector measures the force’s tendency to rotate a solid about O; its magnitude is ||OP|| * ||F|| * sin(theta), and is maximum when the force is perpendicular to the lever arm OP.

Common pitfall

Do not confuse the scalar product (which measures a signed volume) with the vector product (a vector). The sign of the scalar product also indicates the orientation of the triad (u, v, w): positive if direct, negative if indirect. Be sure to distinguish clearly between ‘volume’ (always positive, absolute value) and the mixed product itself (which may be negative).

Quantity	Formula	Nature of the result
Area of the parallelogram
Volume of the parallelepiped	|u . (v ^ w)|	positive number
Cross product	u . (v ^ w)	real number (sign)