Question

5. Determine an equation for the plane that is exactly between the points A(-1, 2, 4) and B(3, 1, -4).

Answer 1

I assume the plane needs to pass through the mid-point between the two points, and perpendicular to the line joining the two points.

First find the mid-point by
M((x1+x2)/2,(y1+y2)/2,(z1+z2)/2)

The normal vector for the plane is just the vector AB, or N(3-(-1),1-2,-4-4)=(4,-1,-8).
The equation for the plane is then:
4(x-xm)-(y-ym)-8(z-zm)=0
where M(xm,ym,zm) as calculated above.