Triple scalar product
The triple scalar product, also known as the scalar triple product or mixed product, is a mathematical operation that combines three vectors to produce a scalar quantity
The triple scalar product, also known as the scalar triple product or mixed product, is a mathematical operation that combines three vectors to produce a scalar quantity. It is denoted as (A ⋅ B) × C or [A, B, C], where A, B, and C are three vectors in three-dimensional space.
The formula for calculating the triple scalar product of three vectors A, B, and C is:
[A, B, C] = A ⋅ (B × C)
Here, the dot product (A ⋅ B) is taken first, followed by the cross product (B × C). The resulting vector from the cross product is then dotted with vector A to obtain the scalar triple product.
Geometrically, the triple scalar product represents the signed volume of the parallelepiped formed by the three vectors A, B, and C. The sign indicates whether the volume is positive, negative, or zero. If the three vectors are coplanar or parallel, the triple scalar product will be zero.
Furthermore, the triple scalar product can also be expressed using determinants:
[A, B, C] = |A B C|
|a1 b1 c1|
|a2 b2 c2|
Where A = (a1, a2), B = (b1, b2), and C = (c1, c2) are two-dimensional vectors. The determinant of this 3×3 matrix yields the value of the triple scalar product.
The triple scalar product has various applications in physics and engineering, particularly in the calculation of moments, forces, and torques. It is a crucial tool in vector algebra and helps analyze the relationships between vectors in three-dimensional space.
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