a vector n of magnitude 8 units is inclined to  x axis at 45degree ,y axis at 60degree and an acute angle with x axis. if a plane passes through  a point (root2,-1,1) and is normal to vector n. find its equation in vector form.

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$$\begin{gathered} \mathrm{Given} \mathrm{that}: \\ \mathrm{\alpha } \mathrm{i}.\mathrm{e} \mathrm{angle} \mathrm{with} \mathrm{x} \mathrm{axis} \mathrm{is} 45 \mathrm{and} \mathrm{\beta } \mathrm{i}.\mathrm{e}. \mathrm{angle} \mathrm{with} \mathrm{y} \mathrm{axis} \mathrm{and} \mathrm{let} \mathrm{\gamma } \mathrm{be} \mathrm{the} \mathrm{angle} \\ \mathrm{with} \mathrm{z} \mathrm{axis}. \\ \mathrm{we} \mathrm{know} \mathrm{that} \\ {\cos }^2\mathrm{\alpha }+{\cos }^2\mathrm{\beta }+{\cos }^2\mathrm{\gamma }=1 \\ {\cos }^{2}45+{\cos }^{2}60+{\cos }^2\mathrm{\gamma }=1 \\ \Rightarrow \frac{1}{2}+\frac{1}{4}+{\cos }^2\mathrm{\gamma }=1 \\ \Rightarrow {\cos }^2\mathrm{\gamma }=1-\frac{1}{2}-\frac{1}{4}=\frac{1}{4} \\ \Rightarrow {\cos }^2\mathrm{\gamma }=\frac{1}{4} \\ \Rightarrow \mathrm{cos\gamma }=\frac{1}{2} \\ \mathrm{Equation} \mathrm{of} \mathrm{normal} \mathrm{vector} \mathrm{is}: \\ \hat{\mathrm{n}}=\mathrm{cos\alpha } \hat{\mathrm{i}}+\mathrm{cos\beta } \hat{\mathrm{j}}+\mathrm{cos\gamma } \hat{\mathrm{k}} \\ =\frac{\mathrm{i}}{\sqrt{2}}+\frac{\hat{\mathrm{j}}}{2}+\frac{\hat{\mathrm{k}}}{2} \\ \mathrm{Given} \mathrm{point} \mathrm{is}: \\ \overrightarrow{\mathrm{a}}=\sqrt{2}\mathrm{i}-\hat{\mathrm{j}}+\hat{\mathrm{k}} \\ \mathrm{Equation} \mathrm{of} \mathrm{plane} \mathrm{will} \mathrm{be}: \\ \overrightarrow{\mathrm{r}}.\hat{\mathrm{n}}=\overrightarrow{\mathrm{a}}.\hat{\mathrm{n}}=\left(\sqrt{2}\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\right).\left(\frac{\hat{\mathrm{i}}}{\sqrt{2}}+\frac{\hat{\mathrm{j}}}{2}+\frac{\hat{\mathrm{k}}}{2}\right)=1-\frac{1}{2}+\frac{1}{2}=1 \\ \Rightarrow \overrightarrow{\mathrm{r}}.\hat{\mathrm{n}}=1 \\ \mathbf{where}\mathbf{ }\hat{\mathbf{n}}\mathbf{=}\frac{\mathbf{i}}{\sqrt{\mathbf{2}}}\mathbf{+}\frac{\hat{\mathbf{j}}}{\mathbf{2}}\mathbf{+}\frac{\hat{\mathbf{k}}}{\mathbf{2}}\mathbf{ }\mathbf{and}\mathbf{ }\overrightarrow{\mathbf{r}}\mathbf{=}\mathbf{x}\hat{\mathbf{i}}\mathbf{+}\mathbf{y}\hat{\mathbf{j}}\mathbf{+}\mathbf{z}\hat{\mathbf{k}} \end{gathered}$$
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