The Cartesian equation of the plane passing through the point \((0,7,-7)\) and containing the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\) is

The plane passes through the point \((0,7,-7)\) contains the line \(\frac{\mathrm{x}+1}{-3}=\frac{\mathrm{y}-3}{2}=\frac{\mathrm{z}+2}{1}\).
\(\therefore\) Required equation of the plane is
\(
\begin{aligned}
& \left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
-1-0 & 3-7 & -2+7 \\
-3 & 2 & 1
\end{array}\right|=0 \Rightarrow\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
-1 & -4 & 5 \\
-3 & 2 & 1
\end{array}\right|=0 \\
& \therefore \hat{\mathrm{i}}(-4-10)-\hat{\mathrm{j}}(-1+15)+\hat{\mathrm{k}}(-2-12)=0
\end{aligned}
\)
Hence Cartesian equation of the plane is \(x+y+z=0\)