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 (x₀, y₀, z₀) 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 (x₁, y₁, z₁), B (x₂, y₂, z₂), C (x₃, y₃, z₃) 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 (x₁, y₁, z₁) lies in this plane, and the vectors u (a₁, b₁, c₁), v (a₂, b₂, c₂) 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 (x₁, y₁, z₁) and parallel to two non-collinear vectors u (a₁, b₁, c₁) and v (a₂, b₂, c₂) is determined by the equation

Equation of a Plane Given Two Points and a Vector

The plane passing through the points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) and parallel to the vector u (a, b, c) is described by the equation

Solved Problems

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.

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:

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:

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:

Solution.

The plane equation in parametric form is written as follows:

where point M has coordinates (x₁, y₁, z₁), the coordinates of vectors u and v are (a₁, b₁, c₁) and (a₂, b₂, c₂), 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