Scalar Triple Product (AS)

The scalar triple product is a concept in vector algebra which is also known as the box product or mixed product. It involves three vectors and delivers a scalar as a result.
Its mathematical representation is [a, b, c] = a . (b x c), where a, b and c are three vectors. The dot denotes the dot product and the cross denotes the cross product.
The scalar triple product is geometrically interpreted as the volume of the parallelepiped formed by the vectors a, b, and c.
If the vectors a, b, and c are co-planar (they lie in the same plane), the volume will be zero, which means the scalar triple product will also be zero.
This leads to an important property: the scalar triple product [a, b, c] = 0 if and only if the vectors a, b and c are linearly dependent or co-planar.
The scalar triple product has the property of being invariate under cyclic permutations. That is, [a, b, c] is equal to [b, c, a] and [c, a, b].
Under anti-cyclic or reverse order permutations, the scalar triple product changes its sign. That means, [a, b, c] is equal to -[c, b, a].
Use proper formulas to calculate the scalar triple product. This includes components of vectors and coordinate geometry.
Understanding the properties of scalar triple product applies to understanding geometric interpretations, proving mathematical identities and solving problems involving three dimensions.
The scalar triple product can also be represented by a determinant of a 3x3 matrix, where each column or row of the matrix represents each vector’s set of parameters. That is: [a, b, c] = |a1 a2 a3 |
|b1 b2 b3 |
|c1 c2 c3 |