[LSN/MM-16001] :: A Quick Reminder of Vector Algebra

In this lesson,  we shall begin with vectors as geometrical objects.
A quick review of a few  vector algebra identities will be presented.
With a choice of coordinate system, vectors are described as objects with three components.
We will present a result  on change in components  of a vector when coordinate axes are changed.

A first exposure to vector algebra; Dot, cross and triple  products.
Components of a vector along coordinate axes.

Need to revise details  about theabove topics?
Watch videos suggested in the video tab.

 Recommended for those who need more details:

1.  Vector algebra
2. Dot and Cross Product of Two vectors  Video 1, Video 2

1. Mathematical equations representing laws of Physics must be scalar or vector equations  (more generally   tensor equations), This means that each term in a  equation must be an object with same transformation properties.Different terms cannot have different transformation properties. For example, a physical law cannot be represented as a vector = a (non zero) scalar quantity.
2. Vectors merely introduced as objects having length and direction, as is done in many  school level text, is not a useful concepts.Vectors when equipped with operation of multiplication by a number, addition by parallelogram law and scalar product becomesuseful concept. The definition of  mathematical concepts such as  Linear Vector Spaces and  Inner Product Spaces has roots in the ordinary vectors as introduced at school level.
3. The concept of vectors is generalized to  tensors and appears in contexts of  many symmetry groups.An example is four vectors and tensors that is required in relativistic theories.   

 For a quick review of vector algebra see
Murphy\cite{Murphy} Ch4;
Griffiths\cite{Griffiths-EM} Ch1;
For use of vectors in Physics see
Feynman Lectures Vol-I\cite{Feynman1} Ch 11.