$\vec{A}$ અને $\vec{B}$ ને કાર્તેઝિય ધટકોના સ્વરૂપમાં નીચે મુજબ લખાય.

$\overrightarrow{ A }= A _{x} \hat{i}+ A _{y} \hat{j}+ A _{z} \hat{k}$

$\overrightarrow{ B }= B _{x} \hat{i}+ B _{y} \hat{j}+ B _{z} \hat{k}$

$\therefore \quad \overrightarrow{ A } \cdot \overrightarrow{ B }=\left( A _{x} \hat{i}+ A _{y} \hat{j}+ A _{z} \hat{k}\right) \cdot\left( B _{x} \hat{i}+ B _{y} \hat{j}+ B _{z} \hat{k}\right)$

$= A _{x} B _{x}(\hat{i} \cdot \hat{i})+ A _{x} B _{y}(\hat{i} \cdot \hat{j})+ A _{x} B _{z}(\hat{i} \cdot \hat{k})$$+ A _{y} B _{x}(\hat{j} \cdot \hat{i})+ A _{y} B _{y}(\hat{j} \cdot \hat{j})+ A _{y} B _{z}(\hat{j} \cdot \hat{k})+ A _{z} B _{x}(\hat{k} \cdot \hat{i})+ A _{z} B _{y}(\hat{k} \cdot \hat{j})+ A _{z} B _{z}(\hat{k} \cdot \hat{k})$

આ સમીકરણમાં $\hat{i} \cdot \hat{i}=\hat{j} \cdot \hat{j}=\hat{k} \cdot \hat{k}=1$ અને

$\hat{i} \cdot \hat{j}=\hat{j} \cdot \hat{i}=0, \hat{j} \cdot \hat{k}=\hat{k} \cdot \hat{j}=0$

$\hat{k} \cdot \hat{i}=\hat{i} \cdot \hat{k}=0$ મૂક્તા,

$\therefore \quad \overrightarrow{ A } \cdot \overrightarrow{ B }= A _{x} B _{x}+ A _{y} B _{y}+ A _{z} B _{z}$ મળે.