Section 11.4

The Cross Product

A vector operation that yields a third vector perpendicular to the first two.

1

Introduction

Unlike the dot product (which gives a scalar), the cross product produces a new vector. Its defining feature is that it points orthogonal (perpendicular) to both input vectors, following the Right Hand Rule. This makes it incredibly useful for finding normal vectors to planes.

Interactive: The Right Hand Rule

is perpendicular to the entire plane containing and .

2

Key Formulas

Determinant Formula

Computed using cofactor expansion:

Magnitude (Area)

Area of parallelogram:

Parallel Vector Test

Vectors are parallel if (sin ):

3

Worked Examples

Example 1: Computing Cross Product (Level 1)

Calculate for and .

Step 1: Set up Determinant

Step 2: Expand Minors

Step 3: Simplify

Example 2: Finding an Orthogonal Vector (Level 2)

Find a unit vector orthogonal to both and .

1. Find Cross Product

(Verify this!)

2. Normalize

Magnitude .
Unit Vector .

Example 3: Area of a Triangle (Level 3)

Find the area of the triangle with vertices .

Strategy: The area of the triangle is area of the parallelogram formed by vectors and .

  • Form Vectors:

  • Cross Product:
  • Area:
4

Practice Quiz

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