1. Let's start by stating the problem: We want to understand the rules for the dot product and cross product of vectors. 2. **Dot product rule:** The dot product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as $$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta$$ where $\theta$ is the angle between the vectors. 3. Important properties of the dot product: - It is commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$. - It distributes over vector addition: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$. 4. **Cross product rule:** The cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is a vector defined as $$\mathbf{a} \times \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \sin \theta \ \mathbf{n}$$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, and $\mathbf{n}$ is a unit vector perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$, following the right-hand rule. 5. Important properties of the cross product: - It is anti-commutative: $\mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a})$. - It distributes over addition: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$. 6. Summary: - Dot product gives a scalar measuring how much two vectors align. - Cross product gives a vector perpendicular to both, measuring the area of the parallelogram they span. Final answer: - Dot product rule: $$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta$$ - Cross product rule: $$\mathbf{a} \times \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \sin \theta \ \mathbf{n}$$