Introduction to Computer Graphics

Appendix B

Properties of Vector Products

This appendix regards the basic properties of vector products and their geometric interpretation.

B.1 Dot Product

The dot product, or scalar product between two vectors is indicated with the symbol ∙ and is defined as:

$\begin{array}{cc}\mathit{\text{a}}\cdot \mathit{\text{b}}={\displaystyle \sum _{\mathit{\text{i}}=1}^{\mathit{\text{n}}}{\mathit{\text{a}}}_1{\mathit{\text{b}}}_1+\cdots +{\mathit{\text{a}}}_{\mathit{\text{n}}}{\mathit{\text{b}}}_{\mathit{\text{n}}}}& \left(\text{B}.1\right)\end{array}$

This equation is a pure algebraic definition where the term vector is intended as a sequence of numbers. When we deal with a geometric interpretation where the vectors are entities characterized by a magnitude and a direction we can write:

$\begin{array}{cc}\mathit{\text{a}}\cdot \mathit{\text{b}}=||\mathit{\text{a}}||||\mathit{\text{b}}||\mathrm{\text{cos}}\mathit{\text{θ}}& \left(\text{B}.2\right)\end{array}$

where θ is the angle formed by the two vectors. Equation (B.2) tells us a few important things.

One of these things is that two non-zero-length vectors are perpendicular to each other if and only ...