10.1 Introduction

Vectors for geometry:

• straight line trajectories;
• surfaces;
• ...

• Dot (scalar) product:
  
  $$\vec a \cdot \vec b = a_x b_x + a_y b_y + a_z b_z = \vert\vec a\vert \vert \vec b\vert \cos \vartheta$$
  
  

• Cross (vector) product:
  
  $$\vert\vec a \times \vec b\vert = \vert\vec a\vert \vert \vec b\vert \sin \vartheta$$
  
  
  and normal to both vectors. Seen from below:
  
  $$\epsffile{vecgeomi1.eps}$$
  
  

• Line through point $P$ parallel to vector $\vec s$:
  
  $$\vec r = \vec r_P + \lambda \vec s$$
  
  
  
  
  $$\epsffile{vecgeomi2.eps}$$
  
  

• Plane through point $P$ normal to vector $\vec n$:
  
  $$\vec n \cdot \vec r = \vec n \cdot \vec r_P$$
  
  
  
  
  $$\epsffile{vecgeomi3.eps}$$
  
  

• Each equation ordinarily reduces the dimensionality by one: 3D (space) $\to$ 2D (plane) $\to$ 1D (line) $\to$ 0D (point) $\to$ nothing.