Find the unit vector orthogonal to the vector 3i+2j+6k and coplanar with the vectors 2i+j+k and i-j+k.

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$$\begin{gathered} 3,2,6 \\ \mathrm{Let} \mathrm{the} \mathrm{desired} \mathrm{vector} \mathrm{be} \overrightarrow{\mathrm{A}}=\mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}} \\ \mathrm{Now} \mathrm{if} \mathrm{three} \mathrm{vectors} \mathrm{are} \mathrm{coplanars} \mathrm{then} \mathrm{one} \mathrm{vector} \mathrm{can} \mathrm{be} \mathrm{written} \mathrm{as} \mathrm{linear} \mathrm{sum} \mathrm{of} \\ \mathrm{other} \mathrm{two}. \\ \Rightarrow \mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}+\mathrm{\lambda }\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\right) \\ \Rightarrow \mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}}=\left(2+\mathrm{\lambda }\right)\hat{\mathrm{i}}+\left(1-\mathrm{\lambda }\right)\hat{\mathrm{j}}+\left(1+\mathrm{\lambda }\right)\hat{\mathrm{k}} \\ \mathrm{Now} \mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}} \mathrm{is} \mathrm{orthogonal} \mathrm{to} 3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+6\hat{\mathrm{k}}, \mathrm{hence} \mathrm{their} \mathrm{dot} \mathrm{product} \mathrm{will} \mathrm{be} 0. \\ \Rightarrow \left(\mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}}\right).\left(3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+6\hat{\mathrm{k}}\right)=0 \\ \Rightarrow \left\{\left(2+\mathrm{\lambda }\right)\hat{\mathrm{i}}+\left(1-\mathrm{\lambda }\right)\hat{\mathrm{j}}+\left(1+\mathrm{\lambda }\right)\right\}.\left(3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+6\hat{\mathrm{k}}\right)=0 \\ \Rightarrow 3\left(2+\mathrm{\lambda }\right)+2\left(1-\mathrm{\lambda }\right)+6\left(1+\mathrm{\lambda }\right)=0 \\ \Rightarrow 6+3\mathrm{\lambda }+2-2\mathrm{\lambda }+6+6\mathrm{\lambda }=0 \\ \Rightarrow 7\mathrm{\lambda }=-14 \\ \Rightarrow \mathrm{\lambda }=-2 \\ \Rightarrow \left(2+\mathrm{\lambda }\right)\hat{\mathrm{i}}+\left(1-\mathrm{\lambda }\right)\hat{\mathrm{j}}+\left(1+\mathrm{\lambda }\right)\hat{\mathrm{k}}=\left(2-2\right)\hat{\mathrm{i}}+\left(1+2\right)\hat{\mathrm{j}}+\left(1-2\right)\hat{\mathrm{k}} \\ \Rightarrow \overrightarrow{\mathrm{A}}=3\hat{\mathrm{j}}-\hat{\mathrm{k}} \\ \Rightarrow \hat{\mathrm{A}}=\frac{3\hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{3^2+{\left(-1\right)}^2}}=\frac{3\hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{9+1}} \\ \Rightarrow \boxed{\hat{\mathbf{A}}\mathbf{=}\frac{\mathbf{3}\hat{\mathbf{j}}\mathbf{-}\hat{\mathbf{k}}}{\sqrt{\mathbf{10}}}} \end{gathered}$$

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