Monotonicity of the Line Function K on a 3D Space

  • Thread starter Asuralm
  • Start date
  • Tags
    Test
In summary, the conversation discusses the relationship between a point and a normal vector in 3D space, and a line function defined by a point and a normalized direction. The question is whether a function defined by the distance between a point on the line and the original point, and the normal vector, varies monotonically. The conversation also suggests using geometry rather than calculus to solve this problem.
  • #1
Asuralm
35
0
Hi all:

Assume in 3D space there is a point [itex]v=[v_x, v_y, v_z][/tex], and a normal vector associate with it as [tex]n=[n_x, n_y, n_z][/tex]. A line function is defined as [tex]u=w+t\cdot l[/tex] where [tex]w=[w_x, w_y, w_z][/tex] is a point, and [tex]l=[l_x, l_y, l_z][/tex] is the normalized direction of the line. l and n are normalized. Assume there is a function defined as:

[tex]
K = \frac{(u-v)\cdot n}{||u-v||^2}
[/tex]

My question is when point u varies on the line, is the function K varies monotonically?

I've tried to compute [tex]\frac{dK}{dt}[/tex], but I can't really see if it's monotone or not, can some one help me please?

Thanks
 
Last edited:
Physics news on Phys.org
  • #2
Hi Asuralm! :smile:

Forget calculus, this is geometry :wink:

Hunt: if u and v represent points U and V, and if the nearest point on L to U is N, what is (u - v).n ? :smile:
 

FAQ: Monotonicity of the Line Function K on a 3D Space

What is the Monotone Test of an equation?

The Monotone Test is a method used to determine the monotonicity of a function or equation. It helps to identify whether the function is increasing, decreasing, or neither over a given interval.

How is the Monotone Test performed?

The Monotone Test involves calculating the first derivative of the function and then determining the sign of the derivative over the given interval. If the derivative is positive, the function is increasing, if it is negative, the function is decreasing, and if it is zero, the function is neither increasing nor decreasing.

Why is the Monotone Test useful?

The Monotone Test is useful because it helps to identify critical points and intervals where the function is either increasing or decreasing. This information can be used to analyze the behavior of the function and make predictions about its properties.

Can the Monotone Test be used for all types of equations?

Yes, the Monotone Test can be used for any type of equation as long as it is differentiable over the given interval. This includes polynomial, exponential, logarithmic, and trigonometric functions.

How is the Monotone Test related to the Concavity Test?

The Monotone Test is closely related to the Concavity Test, as both involve analyzing the behavior of the first derivative of a function. The difference is that the Monotone Test determines the monotonicity of the function, while the Concavity Test determines the concavity of the function. Both tests are important in understanding the overall behavior of a function.

Similar threads

Replies
0
Views
783
Replies
2
Views
1K
Replies
1
Views
1K
Replies
8
Views
781
Replies
1
Views
3K
Replies
5
Views
2K
Back
Top