find an equation for a plane parallel to the xy-plane; does not pass through the origin.
(I know how you would find the equation if it says passes through a certain point. But I'm confused how to do it if it says does not pass through certain point)
the plane
z = k
is parallel to the x-y plane at a distance |k|
The x-y plane is the plane z=0
(which, or course, does pass through the origin)
To find an equation for a plane parallel to the xy-plane that does not pass through the origin, we need to consider that the normal vector of the plane will be perpendicular to the xy-plane. This normal vector will have a zero z-component.
Let's assume that the equation of the plane is of the form Ax + By + Cz + D = 0, where A, B, C are the coefficients of the variables x, y, z, respectively, and D is a constant.
Since the plane is parallel to the xy-plane, it means that the normal vector of the plane is parallel to the z-axis, which means its z-component must be zero. Therefore, we have:
C = 0
Now, to find the values of A, B, and D, we will use the fact that the plane does not pass through the origin. Let's assume the plane passes through the point (x₀, y₀, z₀), which is not the origin. Substituting these coordinates into the equation of the plane, we have:
A*x₀ + B*y₀ + C*z₀ + D = 0
Since C = 0, the equation becomes:
A*x₀ + B*y₀ + D = 0
Now, all we need to do is choose arbitrary values for x₀, y₀, and z₀ to find the values of A, B, and D. For example, let's choose x₀ = 1, y₀ = 1, and z₀ = 1. Substituting these values into the equation, we have:
A*1 + B*1 + D = 0
Simplifying further:
A + B + D = 0
Now, we need to ensure that the chosen values for x₀, y₀, and z₀ do not satisfy this equation. If they do, select different values until they no longer satisfy the equation.
For instance, if we choose A = 1, B = 1, and D = -2, we have:
1 + 1 - 2 = 0
Since the chosen values satisfy the equation, we need to select new values. Let's choose A = 2, B = -1, and D = -3. Substituting these values into the equation, we have:
2 - 1 - 3 = 0
Since the chosen values do not satisfy the equation, we have found a suitable equation for the plane parallel to the xy-plane and not passing through the origin:
2x - y - 3 = 0