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How to Calculate the Maximum Slope of a Plane in Space

Slope is defined as the rise divided by the run. That's a simple operation when you're only dealing with one direction. However, a plane in space has a different slope for every possible direction. A plane may represent a physical object, or it could be a mathematical representation of a physical quantity such as temperature or velocity. Knowing how to find the maximum slope could be useful for finding the quickest or easiest way to achieve whatever the physical quantity represents.

Instructions

- 1
  Rewrite the equation in terms of z. For example, given the equation 2x - y + z - 3 = 0, solve for z: z = 3 - 2x + y.
- 2
  Find the gradient for the function of x and y. The gradient is the vector that points in the direction of the maximum slope. Since the equation is of a plane, the x- and y-components of the vector are the same as the coefficients of x and y. For example, the gradient of f(x,y) = 3 - 2x + y is (-2, 1).
- 3
  Find the magnitude of the gradient with the formula M = sqrt(a^2 + b^2), where a and b are the x- and y-components of the gradient. The magnitude of the gradient is equal to the maximum slope of the plane. For example, M = sqrt((-2)^2 + 1^2) = 2.236.

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