How do you write a plane equation?

To write a plane equation, we first need to understand the components and structure of a plane equation. In three-dimensional space, a plane is a flat surface that extends infinitely in all directions. It can be defined by a point on the plane and a normal vector, which is perpendicular to the plane. The general form of a plane equation is 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0, where 𝑎, 𝑏, and 𝑐 are the components of the normal vector ⃑ 𝑛 = ( 𝑎 , 𝑏 , 𝑐 ). The constant term 𝑑 is determined by substituting the coordinates of a point on the plane into the equation.

Let’s break down the process of writing a plane equation step by step:

1. Determine the normal vector: To write the equation of a plane, we need to find the normal vector ⃑ 𝑛 = ( 𝑎 , 𝑏 , 𝑐 ) that is perpendicular to the plane. This can be done by considering the coefficients of 𝑥, 𝑦, and 𝑧 in the equation. The normal vector represents the direction in which the plane is perpendicular.

2. Find a point on the plane: To fully define the plane, we need a point that lies on it. This point can be given in the problem statement or can be obtained through calculations. If no specific point is given, we can assume any values for 𝑥, 𝑦, and 𝑧 and solve for one of them to find a point that satisfies the equation.

3. Substitute the values into the equation: Once we have the normal vector ⃑ 𝑛 = ( 𝑎 , 𝑏 , 𝑐 ) and a point on the plane (𝑥₀, 𝑦₀, 𝑧₀), we substitute these values into the general plane equation 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0. This allows us to determine the constant term 𝑑, which completes the equation.

4. Simplify the equation if necessary: After substituting the values, we may need to simplify the equation to its simplest form. This can involve rearranging terms, dividing through by common factors, or multiplying through by a constant.

It is important to note that there are infinite plane equations that can represent the same plane. Multiplying all the coefficients (𝑎, 𝑏, 𝑐, 𝑑) by a non-zero constant will yield an equivalent equation. Therefore, it is common practice to write the equation in the simplest form by dividing through by the greatest common divisor of the coefficients.

Let’s consider an example to illustrate the process:

Example: Write the equation of a plane that passes through the point (2, -1, 3) and has a normal vector ⃑ 𝑛 = (1, 2, -1).

1. Determine the normal vector: The given normal vector is ⃑ 𝑛 = (1, 2, -1).

2. Find a point on the plane: The point (2, -1, 3) is given as a point on the plane.

3. Substitute the values into the equation: Substituting 𝑎 = 1, 𝑏 = 2, 𝑐 = -1, 𝑥₀ = 2, 𝑦₀ = -1, and 𝑧₀ = 3 into the general plane equation 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0, we get:

1(𝑥) + 2(𝑦) – 1(𝑧) + 𝑑 = 0.

Substituting the point coordinates, we have:

1(2) + 2(-1) – 1(3) + 𝑑 = 0.
2 – 2 – 3 + 𝑑 = 0.
-3 + 𝑑 = 0.

Solving for 𝑑, we find 𝑑 = 3.

4. Simplify the equation if necessary: The equation 𝑥 + 2𝑦 – 𝑧 + 3 = 0 is already in its simplest form and cannot be further simplified.

Therefore, the equation of the plane that passes through the point (2, -1, 3) and has a normal vector ⃑ 𝑛 = (1, 2, -1) is 𝑥 + 2𝑦 – 𝑧 + 3 = 0.

Writing a plane equation involves understanding the components of a plane and how to determine the normal vector and a point on the plane. By following the steps outlined above and substituting the values into the general form equation, we can easily write the equation of a plane in ℝ³.