# Formulas

## Analytic Geometry # Triple Product

Vectors: $$\mathbf{u},$$ $$\mathbf{v},$$ $$\mathbf{w}$$
Dot product: $$\mathbf{u} \cdot \mathbf{v}$$
Vector product: $$\mathbf{u} \times \mathbf{v}$$
Triple product: $$\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)$$
Coordinates of vectors: $${X_1},$$ $${Y_1},$$ $${Z_1},$$ $${X_2},$$ $${Y_2},$$ $${Z_2},$$ $${X_3},$$ $${Y_3},$$ $${Z_3}$$
Real numbers: $$k$$, $$\lambda$$, $$\mu$$
Volume: $$V$$
1. The scalar triple product (also called the mixed product) of three vectors $$\mathbf{u}$$, $$\mathbf{v}$$ and $$\mathbf{w}$$ is defined as the dot product of the vector $$\mathbf{u}$$ and the cross product of the other two vectors $$\mathbf{v}$$ and $$\mathbf{w}$$:
$$\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right) = \mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right) =$$ $$\mathbf{v} \cdot \left( {\mathbf{w} \times \mathbf{u}} \right) =$$ $$\mathbf{w} \cdot \left( {\mathbf{u} \times \mathbf{v}} \right)$$
2. Circular permutation of the scalar triple product
$$\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right) = \left( {\mathbf{w},\mathbf{u},\mathbf{v}} \right) =$$ $$\left( {\mathbf{v},\mathbf{w},\mathbf{u}} \right) =$$ $$-\left( {\mathbf{v},\mathbf{u},\mathbf{w}} \right) =$$ $$-\left( {\mathbf{w},\mathbf{v},\mathbf{u}} \right) =$$ $$-\left( {\mathbf{u},\mathbf{w},\mathbf{v}} \right)$$
3. Scalar multiplication of the triple product
$$k\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right) =$$ $$k\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)$$
4. Scalar triple product in coordinate form
$$\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right) = \mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right) =$$ $$\left| {\begin{array}{*{20}{c}} {{X_1}} & {{Y_1}} & {{Z_1}}\\ {{X_2}} & {{Y_2}} & {{Z_2}}\\ {{X_3}} & {{Y_3}} & {{Z_3}} \end{array}} \right|,$$
where $$\mathbf{u} = \left( {{X_1},{Y_1},{Z_1}} \right),\;$$ $$\mathbf{v} = \left( {{X_2},{Y_2},{Z_2}} \right),\;$$ $$\mathbf{w} = \left( {{X_3},{Y_3},{Z_3}} \right).$$
5. The volume of a parallelepiped defined by the three vectors $$\mathbf{u}$$, $$\mathbf{v}$$, $$\mathbf{w}$$ is equal to the absolute value of the scalar triple product of these vectors:
$$V = \left| {\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)} \right| =$$ $$\left| {\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right)} \right|$$
6. The volume of a pyramid defined by the three vectors $$\mathbf{u}$$, $$\mathbf{v}$$, $$\mathbf{w}$$ is expressed by the formula
$$V = {\large\frac{1}{6}\normalsize} \left| {\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)} \right| =$$ $${\large\frac{1}{6}\normalsize} \left| {\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right)} \right|$$
7. If the scalar triple product of the vectors $$\mathbf{u}$$, $$\mathbf{v}$$ and $$\mathbf{w}$$ is zero, then the three vectors are linearly dependent (coplanar), i.e. one of the vectors can be represented as a linear combination of the two other vectors:
$$\mathbf{w} = \lambda \mathbf{u} + \mu \mathbf{v},$$
where $$\lambda$$, $$\mu$$ are real numbers.
8. If the scalar triple product of the vectors $$\mathbf{u}$$, $$\mathbf{v}$$ and $$\mathbf{w}$$ is non-zero, then these vectors are linearly independent.
9. The vector triple product of three vectors $$\mathbf{u}$$, $$\mathbf{v}$$ and $$\mathbf{w}$$ is defined as
$$\mathbf{u} \times \left( {\mathbf{v} \times \mathbf{w}} \right) =$$ $$\left( {\mathbf{u} \cdot \mathbf{w}} \right)\mathbf{v} – \left( {\mathbf{u} \cdot \mathbf{v}} \right)\mathbf{w} =$$ $$\left| {\begin{array}{*{20}{c}} \mathbf{v} & \mathbf{w}\\ {\left( {\mathbf{u} \cdot \mathbf{v}} \right)} & {\left( {\mathbf{u} \cdot \mathbf{w}} \right)} \end{array}} \right|$$