Appendix B

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

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:

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

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 ...

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