Vector operations are the building blocks of 3D math. Dot products measure how parallel vectors are, while cross products create new vectors perpendicular to the originals. These tools let us calculate work, torque, and areas in physics and engineering.

Understanding these operations opens up a world of 3D problem-solving. We can find angles between vectors, calculate forces and moments, and determine areas of parallelograms. These concepts are crucial for tackling more complex multivariable calculus problems.

Vector Operations

Concept of dot product

• Definition combines algebraic form $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$ with geometric form $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$
• Geometric interpretation represents scalar projection of one vector onto another, measuring parallelism between vectors (completely parallel, perpendicular)
• Properties include commutative $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$, distributive $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$, and scalar multiplication $(k\mathbf{a}) \cdot \mathbf{b} = k(\mathbf{a} \cdot \mathbf{b})$

Calculation of dot product

• Process involves multiplying corresponding components and summing products
• Angle determination uses formula $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$, solved with inverse cosine
• Special cases include perpendicular vectors (dot product = 0) and parallel vectors (dot product = product of magnitudes)

Cross Product

Definition of cross product

• Algebraic form $\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$ pairs with geometric form $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$
• Geometric interpretation yields vector perpendicular to both inputs, magnitude representing parallelogram area
• Properties encompass anticommutative $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$, distributive $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$, and scalar multiplication $(k\mathbf{a}) \times \mathbf{b} = k(\mathbf{a} \times \mathbf{b})$

Computation of cross product

• Calculation uses determinant method or component-wise approach
• Direction determined by right-hand rule: fingers curl from first to second vector, thumb indicates result
• Magnitude found through $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$ or by calculating resulting vector's magnitude

Applications of vector products

• Practical uses include work calculation $W = \mathbf{F} \cdot \mathbf{d}$, torque $\tau = \mathbf{r} \times \mathbf{F}$, and parallelogram area $A = |\mathbf{a} \times \mathbf{b}|$
• Problem-solving involves identifying appropriate product, simplifying expressions, and combining operations
• Key identities: $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})$ and $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$