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Triple Product

a·(b�c) is a scalar representing the "signed volume" of a parallelepiped (a 6-faced polyhedron all of whose faces are parallelograms lying in pairs of parallel planes) whose dimensions and angles are given by the three vectors a, b, and c.  If the vectors represent three edges of the solid that meet at a point, and they are named in counterclockwise order from the point of view of the interior of the solid (i j k order, if you will), then the volume will be positive.  Otherwise negative.

The triple product is, up to sign, commutative.  That is, any ordering of a, b, and c give a result with the same absolute value, with its sign determined by the order the edges are named.  In detail,

a·(b�c) = b·(c�a) = c·(a�b),

and since a�b = -b�a, it follows that

-a·(b�c) = a·(c�b) = b·(a�c) = c·(b�a)

Matrix Determinant Method

If a, b, c, and d are vectors, then the cross product of a�b and c�d is given by

a�(b�c) = det(a b c) 

where det is the determinant of the square matrix formed by placing the three vectors, one above the other.

Internet references

Mathworld: Scalar Triple Product

Related Pages in this website

Vector Dot Product

Vector Cross Product

Triangle Area Using Vectors, part 1

Triangle Area using Vectors, part 2

Triangle Area using Determinant

Matrix Math

Geometry and Trigonometry, and in particular, the Points and Lines section.


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