Question

Can somebody prove the Dot Product Theorem for me? Can somebody also explain why are a and b are in absolute value signs and provide a visual representation of the theorem if possible?

Answer 1

\[ab = \left| a \right| \left| b \right| \cos \theta \]

Answer 2

\[a dotb=a_{x} b_{x}+a_{y} b_{y}+a_{z} b_{z}\]
This is what Gibbs defined it to be a long time ago.
b(x)=abs(a) cosa(lpha)
b(y)=asina
a(x)=bcosb
a(y)=bsinb
cos(b-a)=cos(theta)=cosbcosa-sinbsina (see euler equation)
\[a_{x} b_{x}+a_{y} b_{y}=acos(a)bcos(b)+asin(a)bsin(b)=abs(a)abs(b)cos(b-a)\]Sorry for the confusing notationing.