Dot and Cross Products

1. **Dot Product:** Yields a scalar mapping projection: 2. **Cross Product:** Yields an orthogonal vector:

Concept Explanation

The dot product measures alignment; it evaluates to zero for orthogonal vectors. The cross product measures rotation/area; it evaluates to zero for parallel vectors and produces a vector perpendicular to the plane formed by the input vectors.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Dot products calculate mechanical work ($W = \mathbf{F} \cdot \mathbf{d}$) and electrical power flow. Cross products compute torque ($\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$) and the electromagnetic Lorentz force.

Example Problem

Compute the cross product of $\mathbf{a} = \langle 1, 0, 2 \rangle$ and $\mathbf{b} = \langle 0, 3, -1 \rangle$.

Solution

Set up the determinant:

$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 2 \\ 0 & 3 & -1 \end{vmatrix}$$

Expand the components:

$$\mathbf{i} (0(-1) - 2(3)) - \mathbf{j} (1(-1) - 2(0)) + \mathbf{k} (1(3) - 0(0))$$

$$= \mathbf{i} (-6) - \mathbf{j} (-1) + \mathbf{k} (3)$$

$$= \langle -6, 1, 3 \rangle$$

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Connections & References

• Parent Concepts: Vectors
• Sub-concepts: Vector Valued Functions
• Course Links: Calc-1 Session 19 | Calc-3 Session 03