Theorem
a^{→} • b^{→} = ||a^{→}|| ||b^{→}|| cos•
Proof
Let us give a modified version of the diagram above.
The three vectors above make the triangle AOB and note that the length of each side is nothing much more than the magnitude of the vector forming that side.
The Law of Cosines defines that,
||a^{→}- b^{→}||^{2} = ||a^{→}||^{2} + ||b^{→}||^{2} - 2 ||a^{→}|| || b^{→}|| cos•
As well by using the properties of dot products we can write the left side as,
||a^{→}- b^{→}||^{2 }= (a^{→} - b^{→}) • (a^{→} - b^{→})
= a^{→}• a^{→} - a^{→}•b^{→} - b^{→}•a^{→} + b^{→}• b^{→}
= || a^{→}||^{2} - 2a^{→} • b^{→} + || b^{→}||^{2}
After that our original equation is,