The value of m for which the straight line 3x-2y+z+3 = 0=4x-3y+4z+1. is parallel to the plane 2x-y+mz-2 = 0 is

Explanation

Given line is the line intersection of planes `3x-2y+z+3=0 and 4x-3y+4z+1=0`.
Now vector along line of intersection is
`" "|{:(hati,,hatj,,hatk),(3,,-2,,1),(4,,-3,,4):}|=-5hati-8hatj-hatk`
Line is parallel to plane `2x-y+mz-2=0`
So plane is perpendicular to vector `-5hati+8hatj-hatk`.
Thus,
`" "(2)(-5)+ (-1)(-8)+ (m)(-1)=0`
or `" "m = -2`