Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?
Strictly speaking the term "dot product" is only used Euclidean space. In other vector spaces the term is "inner product". Of course, any n-dimensional vector space is isomorphic to Rn so the two work out to be "essentially" the same.Also one thing to be aware of is that the algebraic defintion for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k.
and of course here's more info on it from wikipedia:
Project the vector A onto the line through B. Multiply the length of this projection by the length of B. By similar triangles this is the same as projecting the vector B onto the line through A and multiplying the length of the projection by the length of A.Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?