Because the line you are looking for is perpendicular to the other two lines, its direction vector is a multiple of the cross product of the direction vectors of the other two lines.

The direction vector of the first line is [-4, -7, 3] and the direction vector of the...

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Because the line you are looking for is perpendicular to the other two lines, its direction vector is a multiple of the cross product of the direction vectors of the other two lines.

The direction vector of the first line is [-4, -7, 3] and the direction vector of the second line is [3, 2, 4] so the cross product is [-34, 25, 13]. We can use any multiple of this for the direction vector, so let's just use a multiple of 1 (that is, we'll use [-34, 25, 13])

Now use the point given to get the parametric equations:

x=2-34t

y=-5+25t

z=13t

where t is the parameter, and 2, -5 and 0 came from the point that the line passes through.