Triple product
Understanding the Triple Product in Vector Algebra
The triple product in vector algebra can refer to two different products: the scalar triple product and the vector triple product. Both are operations involving three vectors in three-dimensional space.
Scalar Triple Product
The scalar triple product of vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is a scalar quantity obtained by taking the dot product of one vector with the cross product of the other two vectors. It is denoted as $(\vec{a} \cdot (\vec{b} \times \vec{c}))$ and can be interpreted as the volume of the parallelepiped formed by the three vectors.
Formula
The scalar triple product is given by:
$$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ \end{vmatrix} $$
where $\vec{a} = \langle a_1, a_2, a_3 \rangle$, $\vec{b} = \langle b_1, b_2, b_3 \rangle$, and $\vec{c} = \langle c_1, c_2, c_3 \rangle$.
Properties
| Property | Description |
|---|---|
| Anticommutativity | $(\vec{a} \cdot (\vec{b} \times \vec{c})) = -(\vec{a} \cdot (\vec{c} \times \vec{b}))$ |
| Scalar Quantity | The result is a scalar, not a vector. |
| Geometric Interpretation | Represents the volume of the parallelepiped. |
| Sensitivity to Collinearity | If any two vectors are collinear, the product is zero. |
Example
Let $\vec{a} = \langle 1, 2, 3 \rangle$, $\vec{b} = \langle 4, 5, 6 \rangle$, and $\vec{c} = \langle 7, 8, 9 \rangle$. The scalar triple product is:
$$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \ \end{vmatrix} = 0 $$
The determinant is zero, indicating that the vectors are coplanar, and the volume of the parallelepiped is zero.
Vector Triple Product
The vector triple product involves the cross product of one vector with the cross product of the other two vectors. It is denoted as $\vec{a} \times (\vec{b} \times \vec{c})$ and results in a vector that is orthogonal to the original vector $\vec{a}$.
Formula
The vector triple product is given by:
$$ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} $$
Properties
| Property | Description |
|---|---|
| Vector Quantity | The result is a vector, not a scalar. |
| Not Associative | $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ |
| Distributive over Addition | $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$ |
| Orthogonality | The result is orthogonal to $\vec{a}$. |
Example
Let $\vec{a} = \langle 1, 0, 0 \rangle$, $\vec{b} = \langle 0, 1, 0 \rangle$, and $\vec{c} = \langle 0, 0, 1 \rangle$. The vector triple product is:
$$ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} = (1 \cdot 1)\langle 0, 1, 0 \rangle - (1 \cdot 0)\langle 0, 0, 1 \rangle = \langle 0, 1, 0 \rangle $$
The resulting vector is $\langle 0, 1, 0 \rangle$, which is orthogonal to $\vec{a}$.
Conclusion
The triple product in vector algebra is a powerful tool for solving problems involving three-dimensional vectors. The scalar triple product is useful for determining volumes and testing for coplanarity, while the vector triple product helps in finding vectors orthogonal to a given vector. Understanding the properties and applications of both types of triple products is essential for success in vector calculus and related fields.