If $\mathbf{a}=(1,-1,2),\mathbf{b}=(-2,3,5),\mathbf{c}=(2,-2,4)$ and i is the unit vector in the x-direction, then $(\mathbf{a}-2\mathbf{b}+3\mathbf{c})\cdot \mathbf{i}=?$

So, $\begin{array}{l}\mathbf{a}=(1,-1,2),\mathbf{b}=(-2,3,5),\mathbf{c}=(2,-2,4)\\ \mathbf{a}=(1,-1,2)\equiv \mathbf{i}-\mathbf{j}+2\mathbf{k}\\ \mathbf{b}=(-2,3,5)\equiv -2\mathbf{i}+3\mathbf{j}+5\mathbf{k}\end{array}$

and $\mathbf{c}=(2,-2,4)\equiv 2\mathbf{i}-2\mathbf{j}+4\mathbf{k}$

$\begin{array}{l}\Rightarrow \mathbf{a}-2\mathbf{b}+3\mathbf{c}=(\mathbf{i}-\mathbf{j}+2\mathbf{k})-2(-2\mathbf{i}+3\mathbf{j}+5\mathbf{k})+3(2\mathbf{i}-2\mathbf{j}+4\mathbf{k})\\ =11\mathbf{i}-13\mathbf{j}+4\mathbf{k}\text{ and }(\mathbf{a}-2\mathbf{b}+3\mathbf{c})\cdot \mathbf{i}=11\end{array}$