Planes

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Equations for Planes

Based on “Equations for a Plane” in section 1.5.

Key Ideas

The equations are all variations on A(x-x₀) + B(y-y₀) + C(z-z₀) = 0 where A, B, and C are the components of the normal vector and (x₀, y₀, z₀) is any point on the plane.

The concept of a normal vector (i.e., vector perpendicular to the plane) is central to planes, equations for planes, and interactions between planes and other shapes.

Examples of facts about planes that generally follow from the plane equation and the idea of a normal vector include…

• An equation for the distance from a plane to a point, using the plane’s normal vector
• Two intersecting lines define a plane, and can be used to find its normal vector
• Parallel planes have parallel normals
• The intersection of 2 planes is a line

Questions

What’s going on in finding the equation for a plane from the symmetric equations for 2 lines?

What’s going on is mostly getting information for finding a normal vector from the symmetric equations. This amounts to extracting the components of the lines’ direction vectors from the equations, and then taking the cross product of those vectors to find the normal to the plane. Combining that with a point on either line, also extractable from the symmetric equations, lets you write out the standard equation for the plane.

For example, here are symmetric equations for two lines. (In class we actually built these equations from vector equations for the lines, but vector equations are much easier to work with when finding planes than symmetric ones are, making the use of symmetric equations seem like a digression. So imagine that all you have to start with are these symmetric equations.)

Each part of each equation consists of a variable minus the corresponding coordinate of a point on the line, divided by the corresponding component of the direction vector. Based on this, you can read the direction vectors from the equations and take the cross product:

Finally, read the coordinates of a point on one of the lines out of the symmetric equations, and put it and the components of the normal into the standard plane equation:

How do you find the equation for a plane from 3 points in it?

Once again, the key is to find 2 vectors whose cross product you can use as a normal to the plane. This time the vectors are the differences between pairs of points. Once you find the normal, use it with any of the points to construct the plane equation:

Next

Finding the distance from a point to a plane, and other applications of plane equations or normals.

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