vector algebra: dot product and cross product In case of dot product of vectors : $a\cdot b =ab \cos(\alpha)$
and in case of cross product :$a \times b=ab \sin(\alpha)$
question is why $\cos(\alpha)$ in dot product and $\sin(\alpha)$ in cross product.
 A: There are several ways to approach this, depending on how you first define the dot and cross product. 
For the dot product, if you start in $\mathbb R^{2}$ and define the dot product of two vectors $\textbf u=(u_{1},u_{2})$ and $\textbf v=(v_{1},v_{2})$ to be 
$\textbf u\cdot \textbf v=u_{1}v_{1}+u_{2}v_{2}$, then you discover by simple geometry that 
$\textbf u\cdot \textbf v=\vert \textbf u\vert \vert \textbf v\vert \cos \alpha $; $0\leq \alpha \leq \pi$.
On the other hand, if you start with $\textbf u\cdot \textbf v=\vert \textbf u\vert \vert \textbf v\vert \cos \alpha $ as a definition you can show easily that $\textbf u\cdot \textbf v=u_{1}v_{1}+u_{2}v_{2}$. 
So the two definitions give the same result:
i.e. $\tag 1 u_{1}v_{1}+u_{2}v_{2}\Leftrightarrow  \vert u \vert \vert v\vert \cos \alpha$
This extends easily to $3$ dimensions, and thence to higher dimensions, and in fact, provides a nice way to define the angle between two vectors in higher dimensional space. 
In short, the dot product is a function $:\mathbb R^{n}\times \mathbb R^{n}\rightarrow \mathbb R$ with certain properties that ensure that ($1$) is true. 
The cross product, on the other hand, is only defined on vectors in $\mathbb R^{3}$, and gives you back a $\textbf {vector}$. i.e. the cross product is a function $:\mathbb R^{3}\times \mathbb R^{3}\rightarrow \mathbb R^{3}$ 
As in the case of the dot product, there are two ways to go:
You can define $\textbf u\times \textbf v =(\vert \textbf u\vert \vert \textbf v\vert \sin \alpha )\textbf n$ 
where $\textbf n$ is the unique unit vector normal to both $\textbf u$ and $\textbf v$ defined by the right-hand rule, so that immediately, one has $\textbf u\times \textbf v =-\textbf v\times \textbf u $. 
It then follows by calculation, using the additional fact  that $\textbf i\times \textbf j=\textbf k$; $\textbf j\times \textbf k=\textbf i$;$\textbf k\times \textbf i=\textbf j$ that
$\tag 2\textbf u\times \textbf v =(u_{2}v_{3}-u_{3}v_{2})\textbf i+(u_{3}v_{1}-u_{1}v_{3})\textbf j+(u_{1}v_{2}-u_{2}v_{1})\textbf k$.
On the other hand, if you define $\textbf u\times \textbf v=(u_{2}v_{3}-u_{3}v_{2})\textbf i+(u_{3}v_{1}-u_{1}v_{3})\textbf j+(u_{1}v_{2}-u_{2}v_{1})\textbf k$, you discover by calculation that 
$\tag 3 (u_{2}v_{3}-u_{3}v_{2})\textbf i+(u_{3}v_{1}-u_{1}v_{3})\textbf j+(u_{1}v_{2}-u_{2}v_{1})\textbf k=(\vert \textbf u\vert \vert \textbf v\vert \sin \alpha )\textbf n$.
i.e. the two definitions are the same. 
The dot and cross products have simple geometrical interpretations which you can find in any textbook. 
Hope this helps.
