Page 4: Advanced Applications

This page explores more complex applications of vector products in geometry and physics.

Vocabulary: Terms like "Normalenvektor" (normal vector) are introduced to describe vectors perpendicular to a plane.

Example: Detailed calculations show how to find areas of geometric shapes using the kreuzprodukt eigenschaften (cross product properties).

Highlight: The relationship between vector products and geometric properties is emphasized through practical examples.

Introduction to Vector Product

The first page introduces the concept of vector product (kreuzprodukt) and its fundamental properties. The vector product of two vectors produces a third vector that is perpendicular to both input vectors.

Definition: The vector product axb of two vectors a and b in three-dimensional space is a vector perpendicular to both a and b, and thus perpendicular to the plane spanned by these vectors.

Highlight: The resulting vector follows the right-hand rule orientation, where thumb, index finger, and middle finger represent the vectors a, b, and axb respectively.

Key properties include:

• Anticommutativity: bxa = -axb
• Distributivity: (a+b)xc = axc + bxc
• Scalar multiplication: λ(axb) = (λa)xb = ax(λb)

Example: The vector product is commonly used in physics for calculating torque: M = rxF