**Introduction to Geometric Planes**

The equation of a flat geometric plane in Cartesian 3D coordinates is somewhat like the equation of a straight line in Cartesian 2D coordinates.

**Criteria For Uniqueness**

A geometric plane in space can be uniquely defined according to any of the following criteria:

- Three points that do not all lie on the same straight line
- A point in the plane and a vector perpendicular to the plane
- Two intersecting straight lines
- Two parallel straight lines

**General Equation Of Plane**

The simplest equation for a plane is derived on the basis of the second of the foregoing criteria: a point in the plane and a vector normal (perpendicular) to the plane. Figure 9-10 shows a plane *W* in Cartesian three-space, a point *P* = ( *x* _{0}, *y* _{0}, *z* _{0} ) in plane *W* , and a vector ( *a, b, c* ) = *a* **i** + *b* **j** + *c* **k** that is normal to plane *W*. The vector ( *a, b, c* ) in this illustration is shown originating at point *P*, and not at the origin, because this particular plane does not contain the origin (0, 0, 0). The values *x* = *a*, *y* = *b*, and *z* = *c* for the vector are nevertheless based on its standard form.

When these things about a plane are known, we have enough information to uniquely define it and write its equation as follows:

*a* ( *x* − *x* _{0} ) + *b* ( *y* − *y* _{0} ) + *c* ( *z* − *z* _{0} ) = 0

In this form of the equation for a plane, the constants *a*, *b*, and *c* are called the *coefficients*. The above equation can also be written in this form:

*ax* + *by* + *cz* + *d* = 0

where

*d* = −(*ax* _{0} + *by* _{0} + *cz* _{0}) = − *ax* _{0} − *by* _{0} − *cz* _{0}

**Plotting A Plane**

In order to draw a graph of a plane based on its equation, it is sufficient to obtain the points where the plane crosses each of the three coordinate axes. The plane can then be visualized, based on these points.

Not all planes cross all three of the axes in Cartesian *xyz* -space. If a plane is parallel to one of the axes, it does not cross that axis; it may cross one or both of the others. If a plane is parallel to the plane formed by two of the three axes, then it crosses only the axis to which it is not parallel. Any plane in Cartesian three-space must, however, cross at least one of the coordinate axes at some point.

**Geometric Planes Problems and Solutions**

**PROBLEM 1 **

Draw a graph of the plane *W* represented by the following equation:

−2 *x* − 4 *y* + 3 *z* − 12 = 0

**SOLUTION 1**

The *x* -intercept, or the point where the plane *W* intersects the *x* axis, can be found by setting *y* = 0 and *z* = 0 and solving the resulting equation for *x*. Call this point *P* :

Therefore,

*P* = (−6, 0, 0)

The *y* -intercept, or the point where the plane *W* intersects the *y* axis, can be found by setting *x* = 0 and *z* = 0 and solving the resulting equation for *y*. Call this point *Q* :

Therefore,

*Q* = (0, − 3, 0)

The *z* -intercept, or the point where the plane *W* intersects the *z* axis, can be found by setting *x* = 0 and *y* = 0 and solving the resulting equation for *z*. Call this point *R* :

Therefore,

*R* = (0, 0, 4)

These three points are shown in the plot of Fig. 9-11. The plane can be envisioned, based on this data. (The dashed axes are “behind” the plane.)

**PROBLEM 2**

Suppose a plane contains the point (2, −7, 0) and a normal vector to the plane at this point is 3 **i** + 3 **j** + 2 **k**. What is the equation of this plane?

**SOLUTION 2**

The vector 3 **i** + 3 **j** + 2 **k** is equivalent to (*a, b, c*) = (3, 3, 2). We have one point (*x* _{0}, *y* _{0}, *z* _{0}) = (2, −7, 0). Plugging these values into the general formula for the equation of a plane gives us the following:

Practice problems for these concepts can be found at: Vectors And Cartesian Three-Space Practice Test.

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