Find the equation of the plane through the points (3,-2,1) and perpendicular to the vector (4,7,-4).

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$$\begin{gathered} \mathrm{For} \mathrm{a} \mathrm{plane} \mathrm{in} {\mathrm{R}}^3\left(\mathrm{XYZ}\right) \mathrm{with} \overrightarrow{{\mathrm{r}}_0} \mathrm{a} \mathrm{point} \mathrm{that} \mathrm{lies} \mathrm{in} \mathrm{the} \\ \mathrm{plane} \mathrm{and} \overrightarrow{\mathrm{n}} \mathrm{a} \mathrm{vector} \mathrm{normal} \mathrm{to} \mathrm{the} \mathrm{plane}, \mathrm{its} \mathrm{equation} \mathrm{can} \mathrm{be} \mathrm{given} \mathrm{by}: \\ \overrightarrow{\mathrm{n}}.\left(\overrightarrow{\mathrm{r}}-\overrightarrow{{\mathrm{r}}_0}\right)=0 , \mathrm{where} \overrightarrow{\mathrm{r}}=\left(\mathrm{x},\mathrm{y},\mathrm{z}\right) \\ \Rightarrow \overrightarrow{\mathrm{n}}.\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{n}}.\overrightarrow{{\mathrm{r}}_0}=0 \\ \Rightarrow \overrightarrow{\mathrm{n}}.\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{n}}.\overrightarrow{{\mathrm{r}}_0} \\ \mathrm{According} \mathrm{to} \mathrm{question}: \\ \overrightarrow{\mathrm{n}}=4\hat{\mathrm{i}}+7\mathrm{j}-4\mathrm{k} \\ {\mathrm{r}}_0=3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}} \\ \Rightarrow \left(4\hat{\mathrm{i}}+7\mathrm{j}-4\mathrm{k}\right).\left(\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}}\right)=\left(4\hat{\mathrm{i}}+7\mathrm{j}-4\mathrm{k}\right)\left(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}\right) \\ \Rightarrow 4\mathrm{x}+7\mathrm{y}-4\mathrm{z}=12-14-4 \\ \Rightarrow 4\mathrm{x}+7\mathrm{y}-4\mathrm{z}=-6 \\ \Rightarrow \boxed{\mathbf{4}\mathbf{x}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{-}\mathbf{4}\mathbf{z}\mathbf{+}\mathbf{6}\mathbf{=}\mathbf{0}} \end{gathered}$$

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