The formula to calculate the scalar triple product is:
\[ \text{STP} = \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) \]
Where:
The Scalar Triple Product, also known as the Box Product, is a mathematical operation involving three vectors in three-dimensional space. It is denoted as \([a, b, c]\) or \((a \cdot (b \times c))\) and results in a scalar (a single real number). The operation is a combination of the dot product and the cross product. First, the cross product of two vectors (b and c) is calculated, resulting in a new vector. Then, the dot product of the first vector (a) and this new vector is calculated, resulting in a scalar. The scalar triple product has the geometric interpretation as the volume of the parallelepiped spanned by the three vectors. It is zero if and only if the three vectors are coplanar, implying that they lie in the same plane.
Let's assume the following values:
Using the formula:
\[ \text{STP} = (1, 2, 3) \cdot ((4, 5, 6) \times (7, 8, 9)) = (1, 2, 3) \cdot (-3, 6, -3) = 1 \cdot (-3) + 2 \cdot 6 + 3 \cdot (-3) = -3 + 12 - 9 = 0 \]
The Scalar Triple Product is 0.
Let's assume the following values:
Using the formula:
\[ \text{STP} = (2, 3, 4) \cdot ((1, 0, 1) \times (0, 1, 0)) = (2, 3, 4) \cdot (-1, 0, 1) = 2 \cdot (-1) + 3 \cdot 0 + 4 \cdot 1 = -2 + 0 + 4 = 2 \]
The Scalar Triple Product is 2.
