Formulas

Analytic Geometry

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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 parallelepiped
    7. The volume of a pyramid defined by the three vectors \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) is expressed by the formula
      \(V = {\large\frac{1}{6}\normalsize} \left| {\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)} \right| =\) \( {\large\frac{1}{6}\normalsize} \left| {\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right)} \right|\)
    8. The volume of a pyramid
    9. 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.
    10. If the scalar triple product of the vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is non-zero, then these vectors are linearly independent.
    11. The vector triple product of three vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is defined as
      \(\mathbf{u} \times \left( {\mathbf{v} \times \mathbf{w}} \right) =\) \( \left( {\mathbf{u} \cdot \mathbf{w}} \right)\mathbf{v} – \left( {\mathbf{u} \cdot \mathbf{v}} \right)\mathbf{w} =\) \( \left| {\begin{array}{*{20}{c}} \mathbf{v} & \mathbf{w}\\ {\left( {\mathbf{u} \cdot \mathbf{v}} \right)} & {\left( {\mathbf{u} \cdot \mathbf{w}} \right)} \end{array}} \right|\)