Component Vectors

Component Vectors

1. Component Algebra

   • Standard basis vectors
     • Unit vectors
     • Any vector can be written using either components or standard basis vectors
   • 3 space
     • 3 direction cosines
       • all direction angles are les than 180 degrees
   • magnitude and direction are defined just with components
     • magnitude = (x^2+y^2+z^2)^1/2
     • direction tan(theta) = b/a
   • Properties
     • addition of vectors made easy!
     • Scalar multiplocation
     • subtraction of components to find vector between points

2. Dot Product

   • result is scalar
     • no visual representation
   • definition
   • Properties
     • commutative
     • distributive
     • a vector dotted with itself is equal to the magnitude squared
     • do not distribute over dot product
   • Projections
     • formula
       • magnitude
       • full vector
     • the projection onto the axis is the component of the vector
     • Work
       • work is equal to the force dotted with the displacement

3. Cross product

   • result is a vector
   • definition
     • seperate definition for magnitude
   • Properties
     • anti-commutative
     • distributive
     • do not distribute over cross product
     • ******cross before dot****
   • Area and volume of parallelograms and parallelepiped
     • Area for parallelogram
     • Area for a triangle
     • Volume of a parallelepiped
   • Torque
     • rotational force
     • formula