Vector Cross Product

Definition

The vector cross product is a binary operation on two vectors that results in a third vector which is perpendicular to both of the original vectors. It is a fundamental concept in physics, particularly in the study of rotational motion and angular momentum.

5 Must Know Facts For Your Next Test

1. The vector cross product of two vectors $\vec{a}$ and $\vec{b}$ is denoted as $\vec{c} = \vec{a} \times \vec{b}$, and the magnitude of the resulting vector $\vec{c}$ is $|\vec{c}| = |\vec{a}||\vec{b}|\sin\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
2. The direction of the vector cross product $\vec{c}$ is determined by the right-hand rule, where the thumb points in the direction of $\vec{c}$ when the fingers of the right hand are curled from $\vec{a}$ to $\vec{b}$.
3. The vector cross product is anti-commutative, meaning that $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$.
4. The vector cross product is used to calculate angular momentum, which is the product of an object's moment of inertia and its angular velocity.
5. In the context of Newton's Second Law for Rotation, the vector cross product is used to calculate the torque acting on an object, which is the product of the force vector and the position vector.

Review Questions

• Explain how the vector cross product is used to calculate angular momentum.
  • The vector cross product is used to calculate angular momentum, which is a vector quantity that describes the rotational motion of an object. Angular momentum is calculated as the cross product of the position vector $\vec{r}$ and the linear momentum vector $\vec{p}$, such that $\vec{L} = \vec{r} \times \vec{p}$. The magnitude of the angular momentum vector $\vec{L}$ is proportional to the product of the magnitudes of the position and momentum vectors, as well as the sine of the angle between them. The direction of the angular momentum vector is determined by the right-hand rule, and is perpendicular to both the position and momentum vectors.
• Describe how the vector cross product is used to calculate the torque acting on an object in the context of Newton's Second Law for Rotation.
  • In the context of Newton's Second Law for Rotation, the vector cross product is used to calculate the torque acting on an object. Torque is a vector quantity that describes the rotational force acting on an object, and is calculated as the cross product of the position vector $\vec{r}$ and the force vector $\vec{F}$, such that $\vec{\tau} = \vec{r} \times \vec{F}$. The magnitude of the torque vector $\vec{\tau}$ is proportional to the product of the magnitudes of the position and force vectors, as well as the sine of the angle between them. The direction of the torque vector is determined by the right-hand rule, and is perpendicular to both the position and force vectors. This torque vector is then used in the rotational form of Newton's Second Law, $\vec{\tau} = I\vec{\alpha}$, where $I$ is the moment of inertia and $\vec{\alpha}$ is the angular acceleration of the object.
• Analyze the relationship between the vector cross product and the concepts of angular momentum and torque, and explain how they are interconnected in the study of rotational motion.
  • The vector cross product is a fundamental concept that is deeply connected to the study of rotational motion and the related concepts of angular momentum and torque. Angular momentum, which describes the rotational motion of an object, is calculated as the cross product of the position vector and the linear momentum vector, $\vec{L} = \vec{r} \times \vec{p}$. Torque, which describes the rotational force acting on an object, is calculated as the cross product of the position vector and the force vector, $\vec{\tau} = \vec{r} \times \vec{F}$. These two quantities are directly related through the rotational form of Newton's Second Law, $\vec{\tau} = I\vec{\alpha}$, where the torque vector is equal to the product of the moment of inertia $I$ and the angular acceleration vector $\vec{\alpha}$. The vector cross product, by virtue of its anti-commutative property and its ability to generate a vector perpendicular to the original vectors, is the fundamental mathematical operation that allows for the calculation of these rotational quantities and the subsequent analysis of rotational motion in physics.

Related terms

Vector Dot Product: The vector dot product is another binary operation on two vectors that results in a scalar value, representing the magnitude of the projection of one vector onto the other.

Angular Momentum:

Angular momentum is a vector quantity that describes the rotational motion of an object, and is calculated as the cross product of the position vector and the linear momentum vector.

Torque:

Torque is a vector quantity that describes the rotational force acting on an object, and is calculated as the cross product of the position vector and the force vector.