Question Video: Finding the Equation of a Plane Mathematics

Find the equation of the plane π‘₯𝑦.

01:25

Video Transcript

Find the equation of the plane π‘₯𝑦.

If we were to plot this plane on an π‘₯𝑦𝑧-coordinate frame, we would see that it occupies every point in the π‘₯𝑦-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 π‘₯𝑦-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 π‘₯𝑦. 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 π‘₯𝑦-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 π‘₯ 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 π‘₯𝑦.

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