If the lines \(\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}\) intersect, then k has the value

The given equation of lines are \(\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{z}{1}\)
Since the lines intersect,
\(\begin{aligned}
& \quad\left|\begin{array}{ccc}
x_2-x_1 & y_2-y_1 & \mathrm{z}_2-\mathrm{z}_1 \\
\mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 \\
\mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2
\end{array}\right|=0 \\
& \therefore\left|\begin{array}{ccc}
2 & \mathrm{k}+2 & -1 \\
2 & 3 & 4 \\
1 & 2 & 1
\end{array}\right|=0 \\
& \Rightarrow 2(3-8)-(\mathrm{k}+2)(2-4)-1(4-3)=0 \\
& \Rightarrow-10+2 \mathrm{k}+4-1=0 \\
& \Rightarrow 2 \mathrm{k}-7=0 \\
& \Rightarrow \mathrm{k}=\frac{7}{2}
\end{aligned}\)