Question

Find an equation of the plane. the plane through the point (3, 0, 7) and perpendicular to the line x = 5t, y = 2 − t, z = 8 + 4t

Answer 1

Let's write the parametric line

(x,y,z) = (0,2,8) + t(5,-1,4)

(5,-1,4) is called the direction vector of the line.   It's normal to the plane

5x - y + 4z = constant

We get the constant by substituting the point (3,0,7),

5x - y +4z = 5(3) - 0 + 4(7) = 43

5x - y +4z = 43