Question

I think this question is pretty straightforward and I'm making it more complicated than it needs to be.

Find the equation of the vertical plane perpendicular to the y-axis and through the point (2, 3, 4)

I know the x-axis is perpendicular to the y-axis, and in 3D space I know this must be the XZ plane. From here, I get a little lost coming up with the equation.

If the equation of a plane is a(x-x_0) + b(y-y_0) + c(z-z_0) = 0, then my a = 2, b = 3, c = 4. But I feel like that's not right. Do I plug in those values for x, y, z instead?

Answer 1

Ok, the vector equation of a plane is:\[n*(r-r_{0})=0\]where n is a vector normal to the plane, r is the position vector of an arbitrary point in the plane and r0 is the position vector of a given point in the plane. We need a normal vector to solve this. To get a normal vector, we use the given information that our plane is perpendicular to the y-axis. Therefore, our normal vector must be parallel to the y-axis. Any vector parallel to the y-axis will do, so I choose the unit vector <0,1,0> for simplicity. We have:\[<0,1,0>*<x-2,y-3,z-4>=0\]giving,\[y=3\]This makes sense, since the y-coordinate must be constant, and there is no restriction on the values x,z that lie on the plane in 3-space.

Answer 2

This is great, thank you.

Answer 3

no worries man :)