Vector Algebra

If $\overrightarrow{a} = \hat{i} + \hat{j} + 2 \hat{k}$ and $\overrightarrow{b} = 3\hat{i} + 2 \hat{j} - \hat{k}$, find $\left(\overrightarrow{a} + 3 \overrightarrow{b}\right) \cdot \left(2 \overrightarrow{a} - \overrightarrow{b}\right)$

$\begin{aligned} \overrightarrow{a} + 3 \overrightarrow{b} & = \left(\hat{i} + \hat{j} + 2\hat{k}\right) + \left(3 \hat{i} + 2 \hat{j} - \hat{k}\right) \\\\ & = 10 \hat{i} + 7 \hat{j} - \hat{k} \end{aligned}$

$\begin{aligned} 2 \overrightarrow{a} - \overrightarrow{b} & = 2 \left(\hat{i} + \hat{j} + 2 \hat{k}\right) - \left(3 \hat{i} + 2 \hat{j} - \hat{k}\right) \\\\ & = - \hat{i} + 0 \hat{j} +5 \hat{k} \end{aligned}$

$\begin{aligned} \therefore \; \left(\overrightarrow{a} + 3 \overrightarrow{b}\right) \cdot \left(2 \overrightarrow{a} - \overrightarrow{b}\right) & = \left(10 \hat{i} + 7 \hat{j} - \hat{k}\right) \cdot \left(- \hat{i} + 5 \hat{k}\right) \\\\ & = -10 - 5 = - 15 \end{aligned}$