Find the scalar equation for the plane that contains the line r=(2,1,7)+s(0,1,0) and parallel to line?

Question

r=(3,0,4)+t(2,-1,0)

Answer 1

The plane in question contains:

Point P(2,1,7)

Directional vectors:

u = <0,1,0>
v = <2,-1,0>

Take the cross product of u and v to obtain the normal vector n, to the plane.

n = u X v = <0,1,0> X <2,-1,0> = 0i + 0j - 2k
Any multiple will do as well. Divide by -2.
n = k = <0,0,1>

Plug in the point (2,1,7) and obtain the equation of the plane.

0(x - 7) + 0(y -1) + 1(z - 7) = 0
z - 7 = 0