want to understand vector multiplication

[milanmajumder15]

Vector multiplication means dot and cross product. But I actually don't understand it. I understand scalar multiplication, which is an easy form of addition. I can save some time by not continuously adding numbers but multiplying them. But when it comes to vectors I am just memorizing the method of multiplying two vectors in two different ways.
what I am thinking is : 1. F . G = FGcos@ ( but why is that ? )
2. because the dot product means multiplying a vector with its same directional component of another vector. or simply F.G = FxGx+FyGy. ( but why by doing that we are getting a product of multiplication? and why are we calling it a multiplication?)

 

[lev888]

Are you saying that you see how number multiplication is useful but vector multiplication is not?
Both dot product and cross product of vectors are very useful. E.g. "The real world applications of the dot product":

1. F . G = FGcos@ by definition
2. The 2 definitions are equivalent.
Here's one proof: http://www.mit.edu/~hlb/StantonGrant/18.02/details/tex/lec1snip2-dotprod.pdf
Another: