2.4 Products of Vectors

Vector products describe how vectors combine in physics. Scalar products, or dot products, produce a number, while vector products, or cross products, produce another vector with a direction set by the right-hand rule.

These mathematical tools help calculate work, torque, angular momentum, and other quantities where both magnitude and direction matter.

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