$\text{ The value of }\mathrm{m}\text{ for which straight line }3x-2y+z+3=0=4x-3y+4z+1\text{ is }$$\text{ parallel to the plane }2x-y+mz-2=0\text{ is }$

$Let{\overline{n}}_1=\left(3,-2,1\right),{\overline{n}}_2=\left(4,-3,4\right),{\overline{n}}_3=\left(2,-1,m\right)$

$Then{\overline{n}}_1\times {\overline{n}}_{2}isparalleltothelineand\perp tonormal{\overline{n}}_{3}tothegivenplane$

$\Rightarrow \left({\overline{n}}_1\times {\overline{n}}_2\right).{\overline{n}}_3=0or\left[{\overline{n}}_1{\overline{n}}_2{\overline{n}}_3\right]=0\Rightarrow m=-2$