Last week

1
Last week

• Coordinate Systems

Spherical
Cylindrical
Cartesian
Coordinates
Basis vectors
Lamé coefficients
2
Today

• Vector Algebra in Terms of Vector Components

One and the same point P! One and the same vector
E! Different reference directions (unit basis
vectors)
give us different vector components.
3
Ortho-normal Right Handed Triad
Let be one of ,
, or .

• For all three coordinate systems, the coordinate
  induced unit basis is orthogonal and normalized

as well as right handed
4
Scalar Product
5
Vector Product
6
Vector Product as a Determinant
Remember determinants?
Our result for vector product in terms of vector
components
can be written and memorized in a concise form
7
Summary Dot and Cross Product Formulas
8
Vector Component Transformations
An example of coordinate transformations
Given find
An example of vector component transformations
Given find
Vector component is a projection of the vector
onto a reference direction
For example
But how to find , , ,
etc.?
9
Unit Basis Vector Conversions
Directly from the geometry and trigonometric
relationships
For example, table gives us