If you have to find the angle between two vectors, first of all, you should draw them with their beginnings at the common point. Then the smallest of two possible angles is the angle between these vectors (see attached diagram for details).
Now, use the definition of dot product:
[tex]\vec{a}\cdot \vec{b}=|\vec{a}|\cdot |\vec{b}|\cdot \cos \theta_{\vec{a},\vec{b}}.[/tex]
Express the cosine from the previous expression:
[tex]\cos \theta_{\vec{a},\vec{b}}=\dfrac{\vec{a}\cdot \vec{b}}{|\vec{a}|\cdot |\vec{b}|}.[/tex]
Moreover, if [tex]\vec{a}=(a_1,a_2,a_3)[/tex] and [tex]\vec{b}=(b_1,b_2,b_3),[/tex] then
[tex]\cos \theta_{\vec{a},\vec{b}}=\dfrac{a_1\cdot b_1+a_2\cdot b_2+a_3\cdot b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\cdot \sqrt{b_1^2+b_2^2+b_3^2}}.[/tex]
