2.4 Products of Vectors

Vector products are crucial in physics, allowing us to describe complex interactions between forces and objects. They come in two flavors: scalar products (dot products) and vector products (cross products), each with unique properties and applications.

These mathematical tools help us calculate work, torque, and angular momentum. Understanding vector products is key to grasping mechanics, as they provide a powerful way to analyze forces and motion in three-dimensional space.

Vector Products

Scalar vs vector products

• Scalar product (dot product) yields a scalar quantity denoted by $\vec{A} \cdot \vec{B}$ and calculated as $|\vec{A}||\vec{B}|\cos\theta$ where $\theta$ is the angle between vectors
  • Commutative property: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
  • Determines work done by a force $W = \vec{F} \cdot \vec{d}$ and projection of one vector onto another
• Vector product (cross product) yields a vector quantity denoted by $\vec{A} \times \vec{B}$ with magnitude $|\vec{A}||\vec{B}|\sin\theta$
  • Direction perpendicular to the plane containing $\vec{A}$ and $\vec{B}$ determined by the right-hand rule
  • Anti-commutative property: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$
  • Calculates torque $\vec{\tau} = \vec{r} \times \vec{F}$, angular momentum $\vec{L} = \vec{r} \times \vec{p}$, and magnetic force on a moving charge $\vec{F} = q\vec{v} \times \vec{B}$

Vector algebra

• Magnitude of a vector represents its length or size
• Direction of a vector indicates the orientation in space
• Parallelogram rule is used for vector addition
• Distributive property applies: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$

Calculation of scalar products

• Calculate using vector components: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$
• Calculate using magnitudes and angle between vectors: $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$
• Positive scalar product indicates force has a component in the same direction as displacement (work done)
• Negative scalar product indicates force has a component opposite to displacement direction (work against)
• Zero scalar product means force is perpendicular to displacement resulting in no work done
• Determines the component of $\vec{A}$ along the direction of $\vec{B}$ through projection $|\vec{A}|\cos\theta$

Computation of vector products

• Calculate using vector components: $\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$
• Calculate using the determinant of a 3x3 matrix: \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$
• Magnitude $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$ represents the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$
• Direction perpendicular to the plane containing $\vec{A}$ and $\vec{B}$ determined by the right-hand rule (point fingers along $\vec{A}$, curl towards $\vec{B}$, thumb points in direction of $\vec{A} \times \vec{B}$)

Vector products in mechanics

• Torque calculated as $\vec{\tau} = \vec{r} \times \vec{F}$ where $\vec{r}$ is position vector from axis of rotation to force application point and $\vec{F}$ is the applied force
  • Magnitude $|\vec{\tau}| = |\vec{r}||\vec{F}|\sin\theta$ where $\theta$ is angle between $\vec{r}$ and $\vec{F}$
  • Direction perpendicular to plane containing $\vec{r}$ and $\vec{F}$ determined by right-hand rule
• Angular momentum calculated as $\vec{L} = \vec{r} \times \vec{p}$ where $\vec{r}$ is position vector from origin to particle and $\vec{p}$ is linear momentum of particle
  • Magnitude $|\vec{L}| = |\vec{r}||\vec{p}|\sin\theta$ where $\theta$ is angle between $\vec{r}$ and $\vec{p}$
  • Direction perpendicular to plane containing $\vec{r}$ and $\vec{p}$ determined by right-hand rule
• Conservation of angular momentum: total angular momentum of a system remains constant in the absence of external torques