Equation of a Plane Through 3 Points Calculator

What is the Equation of a Plane Through 3 Points?

Imagine you're building a treehouse, and you need to make sure the floor is flat. The equation of a plane through 3 points is like a magic formula that helps you find the perfect flat surface passing through three specific points in space. It's like connecting three stars in the sky to make a flat constellation!

How to Calculate the Equation of a Plane Through 3 Points

Finding this equation is like solving a fun puzzle. We use the coordinates of three points to figure out how our flat surface should be positioned in space. It's similar to adjusting a board until it touches three different tree branches at the same time.

Formula

The magical equation for a plane through three points (x₁, y₁, z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃) is:

\[ a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \]

Where:

• \(a = (y_2 - y_1)(z_3 - z_1) - (z_2 - z_1)(y_3 - y_1)\)
• \(b = (z_2 - z_1)(x_3 - x_1) - (x_2 - x_1)(z_3 - z_1)\)
• \(c = (x_2 - x_1)(y_3 - y_1) - (y_2 - y_1)(x_3 - x_1)\)

Calculation Steps

1. Write down the coordinates of your three points
2. Calculate a, b, and c using the formulas above
3. Plug a, b, c, and (x₁, y₁, z₁) into the main equation
4. Simplify the equation by combining like terms
5. Your final equation is in the form Ax + By + Cz + D = 0

Example and Visual Representation

Let's find the equation of a plane through the points (1, 0, 2), (2, 1, 1), and (-1, 2, 1):

• Calculate a, b, c: a = -3, b = -3, c = 3
• Plug into our equation: -3(x - 1) - 3(y - 0) + 3(z - 2) = 0
• Simplify: -3x - 3y + 3z + 9 = 0
• Final equation: x + y - z - 3 = 0

Here's what this plane looks like:

In this picture, you can see the blue triangle representing our plane, passing through the three red points. It's like a magical floating surface in 3D space, defined by our equation!