Vector Cross Product

Definition

The vector cross product, denoted as $\vec{A} \times \vec{B}$, is a binary operation in vector algebra that produces a vector that is perpendicular to both of the input vectors. The resulting vector has a magnitude equal to the product of the magnitudes of the input vectors and the sine of the angle between them, and its direction is determined by the right-hand rule.

5 Must Know Facts For Your Next Test

1. The vector cross product is used to calculate the magnetic force on a current-carrying conductor in a magnetic field, as described in the topic 22.7 Magnetic Force on a Current-Carrying Conductor.
2. The direction of the vector cross product $\vec{A} \times \vec{B}$ is determined by the right-hand rule, where the thumb points in the direction of the resulting vector when the fingers of the right hand are curled in the direction of the rotation from $\vec{A}$ to $\vec{B}$.
3. The magnitude of the vector cross product $\vec{A} \times \vec{B}$ is equal to the product of the magnitudes of the input vectors $\vec{A}$ and $\vec{B}$ and the sine of the angle between them, $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$.
4. The vector cross product is anti-commutative, meaning that $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.
5. The vector cross product is used to calculate the torque on a current-carrying loop in a magnetic field, as well as the magnetic force on a moving charged particle in a magnetic field.

Review Questions

• Explain how the vector cross product is used to calculate the magnetic force on a current-carrying conductor in a magnetic field.
  • The magnetic force on a current-carrying conductor in a magnetic field is given by the vector cross product of the current vector $\vec{I}$, the length vector $\vec{\ell}$ of the conductor, and the magnetic flux density vector $\vec{B}$. The direction of the magnetic force is determined by the right-hand rule, where the thumb points in the direction of the force when the fingers of the right hand are curled in the direction of the rotation from the current vector to the magnetic field vector. The magnitude of the magnetic force is given by $F = I\ell B\sin\theta$, where $\theta$ is the angle between the current vector and the magnetic field vector.
• Describe the properties of the vector cross product, including its anti-commutative nature and how it differs from the vector dot product.
  • The vector cross product $\vec{A} \times \vec{B}$ is a binary operation in vector algebra that produces a vector that is perpendicular to both $\vec{A}$ and $\vec{B}$. Unlike the vector dot product $\vec{A} \cdot \vec{B}$, which is a scalar value, the vector cross product is a vector quantity with a magnitude and direction. The vector cross product is anti-commutative, meaning that $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$. This is in contrast to the vector dot product, which is commutative, $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$.
• Analyze the role of the vector cross product in the context of magnetic fields and the motion of charged particles, and explain how it can be used to calculate the torque on a current-carrying loop in a magnetic field.
  • The vector cross product is a fundamental concept in the study of electromagnetism and the motion of charged particles in magnetic fields. In the context of 22.7 Magnetic Force on a Current-Carrying Conductor, the vector cross product is used to calculate the magnetic force on a current-carrying conductor in a magnetic field, as the force is given by the cross product of the current vector, the length vector of the conductor, and the magnetic flux density vector. Additionally, the vector cross product is used to calculate the torque on a current-carrying loop in a magnetic field, as the torque is given by the cross product of the position vector of the loop and the magnetic force acting on the loop. The vector cross product is also used to describe the motion of charged particles in a magnetic field, as the force on the particle is given by the cross product of the particle's velocity and the magnetic flux density vector.

Related terms

Vector Dot Product: The vector dot product, denoted as $\vec{A} \cdot \vec{B}$, is a binary operation in vector algebra that produces a scalar value equal to the product of the magnitudes of the input vectors and the cosine of the angle between them.

Right-Hand Rule:

A method for determining the direction of the vector cross product, where the thumb points in the direction of the resulting vector when the fingers of the right hand are curled in the direction of the rotation from the first vector to the second vector.

Magnetic Flux Density:

The magnetic flux density, also known as the magnetic field strength or magnetic induction, is a vector quantity that represents the strength and direction of a magnetic field.