# Plane

## General Equation of a Plane

The general or standard equation of a plane in the Cartesian coordinate system is represented by the linear equation

where (*x*, *y*, *z*) are coordinates of the points belonging to the plane, and *A*, *B*, *C* are real numbers.

## Normal Vector to a Plane

The coordinates of the normal vector **n** (*A*, *B*, *C*) to a plane are the coefficients in the general equation of the plane

## Special Cases of the Equation of a Plane

- If \(A = 0,\) the plane is parallel to the \(x-\)axis;
- If \(B = 0,\) the plane is parallel to the \(y-\)axis;
- If \(C = 0,\) the plane is parallel to the \(z-\)axis;
- If \(D = 0,\) the plane passes through the origin;
- If \(A = B = 0,\) the plane is parallel to the \(xy-\)plane;
- If \(B = C = 0,\) the plane is parallel to the \(yz-\)plane;
- If \(A = C = 0,\) the plane is parallel to the \(xz-\)plane.

## Point Direction Form

where the point *P* (*x*_{0}, *y*_{0}, *z*_{0}) lies in the plane and the vector \(\mathbf{n}\left( {A,B,C} \right)\) is normal to the plane.

## Intercept Form

where the point *a*, *b*, *c* are the intercepts on the *x*, *y*, and *z* axes, respectively.

## Three Points Form

or

where the points *A* (*x*_{1}, *y*_{1}, *z*_{1}), *B* (*x*_{2}, *y*_{2}, *z*_{2}), *C* (*x*_{3}, *y*_{3}, *z*_{3}) lie in the given plane.

## Normal Form

Here *p* is the distance from the origin to the plane, and cos *α*, cos *β*, cos *γ* are the direction cosines of any straight line normal to the plane.

## Parametric Form

where (*x*, *y*, *z*) are the coordinates of any point of the plane, *s* and *t* are parameters, the point *P* (*x*_{1}, *y*_{1}, *z*_{1}) lies in this plane, and the vectors **u** (*a*_{1}, *b*_{1}, *c*_{1}), **v** (*a*_{2}, *b*_{2}, *c*_{2}) are parallel to the plane (see Figure 6 below).

## Equation of a Plane Given a Point and Two Vectors

The plane passing through the point *P* (*x*_{1}, *y*_{1}, *z*_{1}) and parallel to two non-collinear vectors **u** (*a*_{1}, *b*_{1}, *c*_{1}) and **v** (*a*_{2}, *b*_{2}, *c*_{2}) is determined by the equation

## Equation of a Plane Given Two Points and a Vector

The plane passing through the points *P*_{1}(*x*_{1}, *y*_{1}, *z*_{1}) and *P*_{2}(*x*_{2}, *y*_{2}, *z*_{2}) and parallel to the vector **u** (*a*, *b*, *c*) is described by the equation

## Solved Problems

### Example 1.

Find the equation of a plane that cuts off intercepts \(a = 2,\) \(b = -1,\) \(c = 4\) on the coordinate axes.

Solution.

The equation of the plane in intercept form is written as

Substituting the values of *a*, *b* and *c*, we get

Multiply both sides by 4 and move all terms to the left side.

This is the equation of the given plane in general form.

### Example 2.

Find the equation of a plane that passes through the point \(M\left({1,-3,5}\right)\) and cuts off equal intercepts on coordinate axes.

Solution.

Let the plane cut off intercepts of length *a* on the coordinate axes. Then its equation can be written as

Substitute the coordinates of point *M* and define the segment *a*:

Then the equation of the plane looks like this:

or in general form:

### Example 3.

Find the equation of a plane that passes through the point \(P\left({1,2,2}\right)\) and cuts off intercepts \(a = 3,\) \(b = 4\) on the \(x-\)axis and \(y-\)axis, respectively.

Solution.

Let's write down the equation of a plane in intercept form:

We know the values of *a* and *b*, so we have

Let's determine the intercept \(c,\) knowing that the plane passes through the point \(P\left({1,2,2}\right):\)

Then the equation of our plane is written as

It can be easily converted into general form:

### Example 4.

Write the equation of a plane passing through points \(A\left({1,0,-1}\right),\) \(B\left({2,3,-1}\right),\) and \(C\left({1,3,2}\right).\)

Solution.

The equation of a plane passing through three points \(A\left({x_1,y_1,z_1}\right),\) \(B\left({x_2,y_2,z_2}\right),\) \(C\left({x_3,y_3,z_3}\right)\) is written using a determinant and has the form

Let's substitute the known coordinates:

Hence

To calculate this determinant, let's expand it, for example, along the first row:

Simplify this equation:

Dividing both parts by 3 we get the final answer:

### Example 5.

Find the equation of a plane in parametric form and in general form if the plane passes through the point \(M\left({1,2,-4}\right)\) and is parallel to the vectors \(\mathbf{u}\left({-2,3,3}\right)\) and \(\mathbf{v}\left({1,-2,0}\right).\)

Solution.

The plane equation in parametric form is written as follows:

where point *M* has coordinates (*x*_{1}, *y*_{1}, *z*_{1}), the coordinates of vectors **u** and **v** are (*a*_{1}, *b*_{1}, *c*_{1}) and (*a*_{2}, *b*_{2}, *c*_{2}), and *s*, *t* are parameters.

Substitute the coordinates of point *M* and vectors **u**, **v**:

This system represents the equation of the plane in parametric form. Let us now derive the equation of the same plane in general form. Recall that the equation of a plane given a point and two vectors has the following form:

Substituting the coordinates of the point and two vectors we get

Expand the determinant along the third column:

and simplify:

or