​For two vectors ​

​$$\vec{A}=2 \widehat{i}+2 \widehat{j}+3 \widehat{k} \text { and }$$​​

​$$\vec{B}=5 \widehat{i}+2 \widehat{j}+7 \widehat{k}$$, find $$\vec{A} \cdot \vec{B}$$.​

Correct option is D

​$$\begin{aligned}&\vec{A} = 2\hat{i} + 2\hat{j} + 3\hat{k} \implies A_x = 2, \quad A_y = 2, \quad A_z = 3 \\&\vec{B} = 5\hat{i} + 2\hat{j} + 7\hat{k} \implies B_x = 5, \quad B_y = 2, \quad B_z = 7 \\\\&\text{2. Compute the dot product:} \\&\qquad \vec{A} \cdot \vec{B} = (2)(5) + (2)(2) + (3)(7) \\&\qquad \phantom{\vec{A} \cdot \vec{B}} = 10 + 4 + 21 \\&\qquad \phantom{\vec{A} \cdot \vec{B}} = 35\end{aligned}$$​​