Produsul scalar exprimat cu ajutorul coordonatelor

$$\begin{gathered} \mathrm{Fie} (O,\overrightarrow{i},\overrightarrow{j}) \mathrm{un} \mathrm{reper} \mathrm{cartezian} \mathrm{și} \mathrm{vectorii}: \\ \overrightarrow{a}(a_1,a_2) \\ \overrightarrow{b}(b_1,b_2) \\ \overrightarrow{a}=a_1\overrightarrow{i}+a_2\overrightarrow{j} \\ \overrightarrow{b}=b_1\overrightarrow{i}+b_2\overrightarrow{j}. \end{gathered}$$

Expresia analitică a produsului scalar al celor doi vectori este:

$\overrightarrow{a}·\overrightarrow{b}=a_1{b}_1+a_2{b}_2.$

Modulul (norma) unui vector

$\left|\overrightarrow{a}\right|=\sqrt{a_1^2+a_2^2}.$

Cosinusul unghiului a doi vectori 

\(cos\alpha =\frac{a_{1}b_{1}+a_{2}b_{2}}{\sqrt{{a_{1}}^{2}+{a_{2}}^{2}}\cdot \sqrt{{b_{1}}^{2}+{b_{2}}^{2}}}\)