Calculus Planes - General Cartesian Equations

In summary, the general cartesian equations for a vertical plane (parallel to the z-axis), a non-vertical plane, and a horizontal plane (parallel to the x; y-plane) are ax+by-c=0, ax+by+cz=d, and ax+by+cz=d, respectively. Their normal vectors are (a, b, 0), (a, b, c), and (a, b, c), respectively. The cartesian equation for the line of intersection between the plane 5x + y ¡ 2z = 6 (¤) and a general horizontal plane is 5x + y ¡ 2z = d, where d is the constant term in the equation for the horizontal plane. The
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Homework Statement


Write down the general cartesian equation of a vertical plane (parallel to the z-axis), a
non-vertical plane and a horizontal plane (parallel to the x; y-plane) together with their
normal vectors.
Find the cartesian equations of the line of intersection of the plane
5x + y ¡ 2z = 6 (¤)
and a general horizontal plane.
How is the normal to (¤) related to the horizontal normal to the line of intersection?

Homework Equations


ax + by + cz = d

The Attempt at a Solution


It's been a while since I've been back to the Physics Forums so I'm not sure if this is the right section to post this as it is a homework question but I actually have no idea how to solve it.

I've been given a clue that the general cartesian equation of a vertical plane (parallel to the z-axis) is ax+by-c=0, however I do not understand why - I can only guess at best.

I only need help with getting the general cartesian equations, so if anyone could provide some tips or a link to materials about planes it would be great!

Thank you.

EDIT: Nevermind please close, I kinda understand it now.
 
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I've been given a clue that the general cartesian equation of a vertical plane (parallel to the z-axis) is ax+by-c=0, however I do not understand why - I can only guess at best.

z doesn't show in the vertical plane expression because so the "height" z is free to move, go wherever you want.
There is no constraint about the z. It can be whatever you want.
 

FAQ: Calculus Planes - General Cartesian Equations

1. What is a general Cartesian equation in calculus?

A general Cartesian equation in calculus represents a plane in three-dimensional space. It is a mathematical expression that relates the x, y, and z coordinates of points on the plane. It can be written in the form Ax + By + Cz + D = 0, where A, B, and C are constants and x, y, and z are variables.

2. How is a general Cartesian equation different from a specific Cartesian equation?

A general Cartesian equation is a representation of a plane in a more abstract form, while a specific Cartesian equation is a more concrete representation of a specific plane. A specific Cartesian equation may include specific values for the constants A, B, C, and D, making it unique to a particular plane.

3. What information can be gathered from a general Cartesian equation?

A general Cartesian equation contains information about the slope and intercepts of the plane, as well as its orientation in space. By manipulating the equation, one can determine the angle between the plane and each of the coordinate axes, as well as the distance from the origin to the plane.

4. How are general Cartesian equations used in calculus?

In calculus, general Cartesian equations are used to represent and analyze planes in three-dimensional space. They are used to find the slope of a curve on the plane, calculate the area under a curve, and determine the volume of a solid bounded by the plane and other surfaces.

5. What are some real-world applications of general Cartesian equations?

General Cartesian equations are used in various fields such as physics, engineering, and computer graphics. They are used to model and analyze the motion of objects in space, design structures such as buildings and bridges, and create three-dimensional computer-generated images for movies and video games.

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