MHB Find the equation of the plane given a point and two planes

[sawdee]

I've done a question similar to this, however this one has no complete equations i can solve for.

Determine the equation of the plane that passes through (1,3,8) and is perpendicular to the line of intersection of the planes 3x−2z+1=0 and 4x+3y+7=0.

I know to take the cross product of the two normals to get my new direction vector, but I am stuck at that point. What form should this be written in and further, can I use the given point?

 

[Mark44]

I've done a question similar to this, however this one has no complete equations i can solve for.

You're asked to find the equation of the plane with the given point

Determine the equation of the plane that passes through (1,3,8) and is perpendicular to the line of intersection of the planes 3x−2z+1=0 and 4x+3y+7=0.

I know to take the cross product of the two normals to get my new direction vector,

Yes, that will work.

but I am stuck at that point. What form should this be written in and further, can I use the given point?

There are several different forms of the equation of a plane, so unless the grader is especially particular, it doesn't matter much what form you use.
You have to use the given point in order to get a unique plane. If N is the direction of the intersection of the given planes (and hence is normal to the plane you want), then the equation of the plane is $n_1(x - 1) + n_2(y - 3) + n_3(z -8) = 0$, where $\vec N = <n_1, n_2, n_3>$.