# Equation of the Plane: Solving for Unknowns | Homework Help

• -EquinoX-
In summary, The equation of the given plane is z = -2y + 2, with x being arbitrary. The slope of the line in the y-z plane that generates the plane is -2.

## Homework Statement

http://img245.imageshack.us/img245/7428/equation.th.jpg [Broken]

## The Attempt at a Solution

I feel really stupid that I can't find the equation of the following... is the y slope here 2?

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I think so, ie. one unit across two up

but it's not z = f(x,y) = 2y right?? what am I missing here

-EquinoX- said:

## Homework Statement

http://img245.imageshack.us/img245/7428/equation.th.jpg [Broken]

## The Attempt at a Solution

I feel really stupid that I can't find the equation of the following... is the y slope here 2?
Don't know what you mean "y slope." This plane appears to by generated by a line in the y-z plane whose slope (dz/dy) is -2. The equation of that line is z = -2y + 2. x is arbitrary.

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## What is the equation of a plane?

The equation of a plane is a mathematical representation of a two-dimensional flat surface in a three-dimensional space. It is typically written in the form of Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the x, y, and z variables and D is a constant term.

## How do you find the equation of a plane given three points?

To find the equation of a plane given three points, you can use the formula (x-x1)(y-y2)(z-z3) + (y-y1)(z-z2)(x-x3) + (z-z1)(x-x2)(y-y3) = (x-x1)(y-y3)(z-z2) + (y-y1)(z-z3)(x-x2) + (z-z1)(x-x3)(y-y2), where (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) are the coordinates of the three points.

## What are the different forms of the equation of a plane?

Aside from the standard form (Ax + By + Cz + D = 0), the equation of a plane can also be written in vector form as r ⋅ n = p, where r is a position vector, n is a normal vector to the plane, and p is a constant. It can also be expressed in parametric form as x = x0 + at, y = y0 + bt, and z = z0 + ct, where (x0, y0, z0) is a point on the plane and a, b, and c are the direction ratios of a vector parallel to the plane.

## What is the importance of the equation of a plane in geometry and physics?

The equation of a plane plays a crucial role in both geometry and physics. In geometry, it is used to represent and analyze two-dimensional figures and shapes such as triangles, rectangles, and circles. In physics, the equation of a plane is used to describe the motion of objects in three-dimensional space, such as the trajectory of a projectile or the path of a moving particle.

## Can the equation of a plane be used to find the distance between a point and a plane?

Yes, the equation of a plane can be used to find the distance between a point and a plane. This can be done using the formula d = |Ax0 + By0 + Cz0 + D| / √(A2 + B2 + C2), where (x0, y0, z0) is the coordinates of the point and A, B, and C are the coefficients of the plane's equation. This formula is derived from the fact that the shortest distance between a point and a plane is along a line perpendicular to the plane, which can be represented by the normal vector (A, B, C).