The Cartesian equation of the plane \(\bar{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2 \hat{j}+3 \hat{k})\) is

\(
\begin{aligned}
& \overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \\
& \overline{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}, \overline{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{B}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}
\end{aligned}
\)
\(\overline{\mathrm{n}}\) is \(\perp\) er \(\overline{\mathrm{A}}\) and \(\overline{\mathrm{B}}\)
\(
\begin{aligned}
& \therefore \overline{\mathrm{n}}=\overline{\mathrm{A}} \times \overline{\mathrm{B}}=\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
1 & 1 & 1 \\
1 & -2 & 3
\end{array}\right| \\
& =\hat{\mathrm{i}}(3+2)-\hat{\mathrm{j}}(3-1)+\hat{\mathrm{k}}(-2-1)=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \\
& \overline{\mathrm{a}} \cdot \overline{\mathrm{n}}=\mathrm{d}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}) \cdot(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=5+2=7 \\
& \therefore \overline{\mathrm{r}} \cdot(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7 \\
& \Rightarrow \text { Cartesian equation is } 5 \mathrm{x}-2 \mathrm{y}-3 \mathrm{z}-7=0
\end{aligned}
\)