Question

Give the standard equation of a plane in space. Describe what is required to find this equation.

Short answer

The standard equation of a plane in space is given by \(Ax + By + Cz = D\). To find this equation, you need a point lying on the plane and the normal vector to the plane.

Step-by-step solution

Understanding the Standard Equation of a Plane

The standard equation of a plane in the 3-dimensional space is generally given by \(Ax + By + Cz = D\). In this equation, A, B, and C are the coefficients which can be interpreted as the normal vector \(\vec{n} = \langle A, B, C \rangle\) to the plane. D is the constant term.

Requirements for the Equation

To find the equation of a plane, you need the following: 1. A point that lies on the plane given by coordinates (x0, y0, z0). 2. The normal vector to the plane indicated by the coefficients A, B, and C.

Formulating the Equation

Once you have the required point and normal vector, the equation can be structured as follows: \(A(x - x0) + B(y - y0) + C(z - z0) = 0\). This equation can be further simplified to the standard form by multiplying out the terms on the left side and simplifying. The end form will ultimately result in \(Ax + By + Cz = D\).