Vectors: \(\mathbf{u},\) \(\mathbf{v},\) \(\mathbf{w}\)
Dot product: \(\mathbf{u} \cdot \mathbf{v}\)
Vector product: \(\mathbf{u} \times \mathbf{v}\)
Triple product: \(\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)\)
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: \(V\)
Real numbers: \(k\), \(\lambda\), \(\mu\)
Volume: \(V\)
- 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 \(\mathbf{w}\):
\(\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right) = \mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right) =\) \( \mathbf{v} \cdot \left( {\mathbf{w} \times \mathbf{u}} \right) =\) \( \mathbf{w} \cdot \left( {\mathbf{u} \times \mathbf{v}} \right)\) - Circular permutation of the scalar triple product
\(\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right) = \left( {\mathbf{w},\mathbf{u},\mathbf{v}} \right) =\) \( \left( {\mathbf{v},\mathbf{w},\mathbf{u}} \right) =\) \( -\left( {\mathbf{v},\mathbf{u},\mathbf{w}} \right) =\) \( -\left( {\mathbf{w},\mathbf{v},\mathbf{u}} \right) =\) \( -\left( {\mathbf{u},\mathbf{w},\mathbf{v}} \right)\) - Scalar multiplication of the triple product
\(k\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right) =\) \( k\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)\) - 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 \(\mathbf{u} = \left( {{X_1},{Y_1},{Z_1}} \right),\;\) \(\mathbf{v} = \left( {{X_2},{Y_2},{Z_2}} \right),\;\) \(\mathbf{w} = \left( {{X_3},{Y_3},{Z_3}} \right).\) - 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 pyramid defined by the three vectors \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) is expressed by the formula
\(V = {\large\frac{1}{6}\normalsize} \left| {\left( {\mathbf{u},\mathbf{v},\mathbf{w}} \right)} \right| =\) \( {\large\frac{1}{6}\normalsize} \left| {\mathbf{u} \cdot \left( {\mathbf{v} \times \mathbf{w}} \right)} \right|\) - 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. - If the scalar triple product of the vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is non-zero, then these vectors are linearly independent.
- The vector triple product of three vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is defined as
\(\mathbf{u} \times \left( {\mathbf{v} \times \mathbf{w}} \right) =\) \( \left( {\mathbf{u} \cdot \mathbf{w}} \right)\mathbf{v} – \left( {\mathbf{u} \cdot \mathbf{v}} \right)\mathbf{w} =\) \( \left| {\begin{array}{*{20}{c}} \mathbf{v} & \mathbf{w}\\ {\left( {\mathbf{u} \cdot \mathbf{v}} \right)} & {\left( {\mathbf{u} \cdot \mathbf{w}} \right)} \end{array}} \right|\)
