Finding the Equation of a Plane
In this article, we will learn how to find the equation of a plane that passes through the origin and two given points.
Step 1: Understand the problem
We are given the following information:
- The plane passes through the origin (0, 0, 0)
- The plane also passes through the points (2, -4, 6) and (5, 1, 3)
To find the equation of the plane, we need to determine the normal vector of the plane, which is perpendicular to the plane.
Step 2: Find the normal vector
The normal vector of the plane can be found by taking the cross product of two vectors that lie on the plane.
Let's define two vectors:
- Vector 1: from the origin to the point (2, -4, 6)
- Vector 2: from the origin to the point (5, 1, 3)
Vector 1 = (2, -4, 6) Vector 2 = (5, 1, 3)
The normal vector is the cross product of these two vectors: Normal vector = Vector 1 × Vector 2 = (2, -4, 6) × (5, 1, 3) = (-27, -15, 17)
Step 3: Write the equation of the plane
The equation of a plane in 3D space can be written in the form: Ax + By + Cz + D = 0
where (A, B, C) is the normal vector, and D is a constant.
Since the plane passes through the origin, we know that the constant D is equal to 0.
Therefore, the equation of the plane is: -27x - 15y + 17z = 0
This is the equation of the plane that passes through the origin and the points (2, -4, 6) and (5, 1, 3).