A flat plane on a 3D cartesian grid is described by the equation z = 2.73x - 1.51y - 1.99
Consider the positive x-axis to be zero degrees, the positive y-axis to be 90 degrees, the negative x-axis to be 180 degrees etc.
Too dec places please. thanks again guys
Direction of steepest descent (degrees) | Absolute angle of steepest descent (degrees) |
---|---|
If you were standing at (x,y) = (1.3,4.8), in which horizontal direction is the steepest slope DOWNWARDS ?
In a flat plane: The gradient is independent from the place ( co-ordinates ).
The gradient points in every point of the space in the direction of the strongest increase.
f(x,y) z = 2.73x - 1.51y - 1.99
$$\dfrac{\partial f(x,y)}{\partial x} = 2.73 \quad and \quad
\dfrac{\partial f(x,y)}{\partial y} = -1.51\\
Gradient =
\left(
\begin{array}{r}
2.73 \\
-1.51
\end{array}
\right)\\$$
horizontal direction is the steepest slope UPWARDS =
$$\tan^{-1}(\frac{-1.51}{2.73}) = -28\ensurement{^{\circ}}.9475759928 + 360 \ensurement{^{\circ}} =331\ensurement{^{\circ}}.052424007$$
horizontal direction is the steepest slope DOWNWARDS = $$331\ensurement{^{\circ}}
.052424007 - 180\ensurement{^{\circ}} = 151\ensurement{^{\circ}}.052424007$$
Consider the positive x-axis to be zero degrees, the positive y-axis to be 90 degrees, the negative x-axis to be 180 degrees etc.
In that direction, what is the absolute angle between the surface and the horizontal plane?
slope: $$\tan^{-1}( \sqrt{1.51^2+2.73^2 } ) = \tan^{-1}(3.11977563296)
= 72\ensurement{^{\circ}}.2274815207$$
The rate of change of z with respect to x is just 2.73, which is positive, so the slope in the positive x-direction is up, and in the negative x-direction is down.
The rate of change of z with respect to y is -1.51 which is negative, so the slope in the positive y-direction is down and in the negative y-direction is up
Go with heureka's answer - I've changed my mind on this twice already!!
If you were standing at (x,y) = (1.3,4.8), in which horizontal direction is the steepest slope DOWNWARDS ?
In a flat plane: The gradient is independent from the place ( co-ordinates ).
The gradient points in every point of the space in the direction of the strongest increase.
f(x,y) z = 2.73x - 1.51y - 1.99
$$\dfrac{\partial f(x,y)}{\partial x} = 2.73 \quad and \quad
\dfrac{\partial f(x,y)}{\partial y} = -1.51\\
Gradient =
\left(
\begin{array}{r}
2.73 \\
-1.51
\end{array}
\right)\\$$
horizontal direction is the steepest slope UPWARDS =
$$\tan^{-1}(\frac{-1.51}{2.73}) = -28\ensurement{^{\circ}}.9475759928 + 360 \ensurement{^{\circ}} =331\ensurement{^{\circ}}.052424007$$
horizontal direction is the steepest slope DOWNWARDS = $$331\ensurement{^{\circ}}
.052424007 - 180\ensurement{^{\circ}} = 151\ensurement{^{\circ}}.052424007$$
Consider the positive x-axis to be zero degrees, the positive y-axis to be 90 degrees, the negative x-axis to be 180 degrees etc.
In that direction, what is the absolute angle between the surface and the horizontal plane?
slope: $$\tan^{-1}( \sqrt{1.51^2+2.73^2 } ) = \tan^{-1}(3.11977563296)
= 72\ensurement{^{\circ}}.2274815207$$