Purpose of Dot/Cross Product currently I am working on Linear Algebra in University. I have a keen interest in mathematics and I also know that it can become frustrating at times. Right know I am frustrated and I seek for help in understanding the following:


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*I know how to evaluate the dot/cross product and projection, but my biggest problem is when it comes to the exercises. I can't figure out what the results on the dot/cross product represent. My textbook is full with formal proofs, but for the sake of usability is does not a great job. I have my exam by the end of December and I need some advice so I can continue my studies and actually understand the more advanced topics such as Euclidean Vector Spaces. I apologize if this is not the usually question, but I am desperate for help. Thank you very much for your time and effort.


-Daniel
 A: Geometrically Speaking:
The dot product is vital for measuring lengths and angles. First ${\bf u} \cdot {\bf u} = ||{\bf u}||^2$ and second, assuming ${\bf u}$ and ${\bf v}$ are non-zero, we have ${\bf u} \cdot {\bf v} = 0$ if and only if ${\bf u}$ and ${\bf v}$ are perpendicular.
The cross product of two non-zero, non-parallel vectors gives a third vector that is perpendicular to the first two. Thus: ${\bf u} \perp ({\bf u} \times {\bf v}) \perp {\bf v}.$
Moreover, the dot product and the cross product are related by the triple scalar product:
$$[{\bf u},{\bf v},{\bf w}] = ({\bf u} \times {\bf v})\cdot {\bf w}$$
The triple scalar product is the volume of the parallelepiped spanned by ${\bf u}$, ${\bf v}$ and ${\bf w}$.
Abstractly Speaking:
The dot product can be used to show that a vector space $V$ is isomorphic to its dual vector space $V^*$. Recall that $V^*$ consists of all linear maps $f : V \to \mathbb{R}.$ The isomorphism $\phi : V \to V^*$ is given by $\phi : {\bf u} \mapsto f_{{\bf u}}$ where $f_{{\bf u}}({\bf v}) = {\bf u} \cdot {\bf v}$ for all ${\bf v}$ in $V$.
