Triple Product

Definition

The triple product, also known as the scalar triple product, is a mathematical operation that combines three vectors in three-dimensional space to produce a scalar value. It represents the volume of the parallelepiped formed by the three vectors and is a useful tool in various applications, including physics and geometry.

5 Must Know Facts For Your Next Test

1. The triple product of three vectors $\vec{a}, \vec{b}, \vec{c}$ is denoted as $\vec{a} \cdot (\vec{b} \times \vec{c})$, which is a scalar value.
2. The triple product is equal to the volume of the parallelepiped formed by the three vectors, with the sign of the result indicating the orientation of the parallelepiped.
3. The triple product satisfies the property of multilinearity, meaning it is linear in each of its three arguments.
4. The triple product can be used to determine whether three vectors are linearly independent, as the triple product will be non-zero if and only if the vectors are linearly independent.
5. The triple product has applications in physics, such as in the calculation of the angular momentum of a rigid body and the evaluation of the work done by a force in a closed path.

Review Questions

• Explain the geometric interpretation of the triple product and how it relates to the volume of a parallelepiped.
  • The triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ represents the volume of the parallelepiped formed by the three vectors $\vec{a}, \vec{b}, \vec{c}$. The magnitude of the triple product gives the volume of the parallelepiped, while the sign of the result indicates the orientation of the parallelepiped, with a positive sign indicating a right-handed orientation and a negative sign indicating a left-handed orientation. This geometric interpretation makes the triple product a useful tool in various applications, such as in the analysis of rigid body motion and the calculation of angular momentum.
• Describe the multilinearity property of the triple product and explain how it can be used to determine the linear independence of three vectors.
  • The triple product satisfies the property of multilinearity, meaning that it is linear in each of its three arguments. This implies that the triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ is zero if and only if at least one of the vectors is a linear combination of the other two. Therefore, the triple product can be used to determine whether three vectors are linearly independent. If the triple product is non-zero, then the three vectors are linearly independent; if the triple product is zero, then the vectors are linearly dependent.
• Discuss the applications of the triple product in physics, specifically in the calculation of angular momentum and the evaluation of work done by a force in a closed path.
  • The triple product has important applications in physics, particularly in the calculation of angular momentum and the evaluation of work done by a force in a closed path. The angular momentum of a rigid body about a point is given by the triple product of the position vector from the point to a point on the body and the linear momentum of that point. Additionally, the work done by a force in a closed path is equal to the negative of the triple product of the force vector and the position vector, integrated over the closed path. These applications highlight the usefulness of the triple product in various areas of physics, where it provides a concise and powerful way to represent and manipulate vector quantities.

Related terms

Scalar Product: The scalar product, also called the dot product, is a binary operation that takes two vectors and produces a scalar value, which represents the magnitude of one vector projected onto the other vector.

Vector Product: The vector product, also called the cross product, is a binary operation that takes two vectors and produces a third vector that is perpendicular to both of the original vectors.

Parallelepiped: A parallelepiped is a three-dimensional geometric shape formed by six parallelogram faces. The volume of a parallelepiped is given by the triple product of its three defining vectors.