Video Transcript
Find the equation of the plane
𝑥𝑦.
If we were to plot this plane on an
𝑥𝑦𝑧-coordinate frame, we would see that it occupies every point in the
𝑥𝑦-plane. To find the equation of this plane,
we’ll need to know a vector that is normal to it as well as a point that lies in
it. As far as a point that lies in the
plane, we can pick the origin. This lies in the 𝑥𝑦-plane and has
coordinates zero, zero, zero. And then what about a vector that’s
normal to this plane? We can see that a vector that
points along the 𝑧-axis would be perpendicular to the plane 𝑥𝑦. This tells us that a vector with
components zero, zero, one that starts at the origin and points one unit in the
positive 𝑧-direction is normal to our 𝑥𝑦-plane.
Now, generally speaking, the
equation of a plane can be given by a vector normal to it and a vector to a point
that lies in it. That normal vector dotted with a
vector to a general point in the plane is equal to the normal vector dotted with a
vector to a known point. In our scenario, we have a normal
vector with components zero, zero, one and a point in the plane with coordinates
zero, zero, zero. When we compute these two dot
products, we find that zero 𝑥 plus zero 𝑦 plus one 𝑧 equals zero plus zero plus
zero. Or, simplifying both sides, 𝑧
equals zero. This is the equation of the plane
𝑥𝑦.
