Formulas and Tables

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:
    \(V = \left| {\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)} \right| =\) \( \left| {\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right)} \right|\)
The volume of a parallelepiped
  1. The volume of a pyramid defined by the three vectors \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) is expressed by the formula
    \(V = {\large\frac{1}{6}\normalsize} \left| {\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)} \right| =\) \( {\large\frac{1}{6}\normalsize} \left| {\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right)} \right|\)
The volume of a pyramid
  1. 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.
  2. If the scalar triple product of the vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is non-zero, then these vectors are linearly independent.
  3. The vector triple product of three vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is defined as
    \(\mathbf{u} \times \left( {\mathbf{v} \times \mathbf{w}} \right) =\) \( \left( {\mathbf{u} \cdot \mathbf{w}} \right)\mathbf{v} – \left( {\mathbf{u} \cdot \mathbf{v}} \right)\mathbf{w} =\) \( \left| {\begin{array}{*{20}{c}}
    \mathbf{v} & \mathbf{w}\\
    {\left( {\mathbf{u} \cdot \mathbf{v}} \right)} & {\left( {\mathbf{u} \cdot \mathbf{w}} \right)}
    \end{array}} \right|\)