If I want to divide vectors and produce a scalar quotient can I go as follows: [tex]\frac{\vec{u}}{\vec{v}} \cdot \frac{\vec{v}}{\vec{v}}[/tex] i.e. compute the dot products and then divide
Well you do not have many options. In order to have a scalar product , you need either a)two vectors or b)two numbers or c)a vector and a number that you multiply... The quotient of two vectors is NOT a vector nor a number. The only thing that you can do is first calculate the scalar product in the numerator and then the scalar product in the denominator. This yields two numbers (ie scalars) that you can devide... marlon
In other words, [tex]\frac{\vec{u}\cdot \vec{u}}{\vec{v}\cdot\vec{v}}[/tex]. In fact, I might be inclined to take the square root of that: [tex]\sqrt{\frac{\vec{u}\cdot \vec{u}}{\vec{v}\cdot{\vec{v}}}[/tex]. so that you are really dividing the lengths of the two vectors. Of course, that will not have very nice properties. Division of vectors is not normally defined. What are you doing this for?