Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems
  Example 1 :
  Express an arbitray vector $\vec A$ as a linear combination of three non-coplanar vectors $\vec a, \vec b $ and $\vec c$.
  Solution :
  Let $\vec A = \alpha\vec a + \beta\vec b + \gamma\vec c$. Since the cross product $\vec b\times\vec c$ is perpendicular to both $\vec b$ and $\vec c$, its dot product with both vectors is zero. Taking the dot product of $\vec A$ with $\vec b\times\vec c$, we have
 
 
which gives
 
  The coefficients $\beta$ and $\gamma$ may be found in a similar fashion.

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