Introduction

Vectors:

• Statics and Dynamics
• Physics
• Geometry
• Computer solutions
• ...

Properties:

• The vector above as a list of numbers:

  

  If the list is written vertically, the vector is called a column vector; if it is written horizontally, it is a row vector. An n-dimensional row vector is equivalent to a size matrix; an n-dimensional column vector to a matrix,
• Components of the vector above: r₁=r_x=x₁=x=4, r₂=r_y=x₂=y=2.
• Addition of vectors:

  

  (4,2)+(1,3)=(4+1,2+3)=(5,5).
• Multiplication of a vector by a scalar:

  

  .
• Length or norm of a vector:

  

  • Definition:

    

  • Unit vectors: Unit vectors have length one.
  • Distance: The distance between two points and is by definition :

    

• Dot (scalar) product:

  

  • Definition:

    

  • Orthogonality: If the dot product is zero, the vectors are by definition orthogonal to each other.
  • Length: .
• Projection:

  

  The magnitude of the (orthogonal) component (or coordinate) of in the direction of is:

  

  The projection of onto is