Identify these surfaces- quick vector question

In summary, part i is an ordinary plane and part ii is a line that passes through the origin and has a distance of 1 to every point on it.
  • #1
Lucy Yeats
117
0

Homework Statement


Identify the following surfaces:
i) r.u=L
ii) r.u=mlrl for -1[itex]\leq[/itex]m[itex]\leq[/itex]1
where k, L, m are fixed scalars and u is a fixed unit vector.


Homework Equations





The Attempt at a Solution


The first one is in the same form as the equation of a plane, but u is not necessarily the normal, so I'm confused. For the second one, I have no idea.
 
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  • #2
Is part i just an ordinary plane?
 
  • #3
Hi Lucy Yeats! :smile:

Lucy Yeats said:
Is part i just an ordinary plane?

Yes.

r.u is the distance of point r to the surface perpendicular to u through the origin.

With r.u=L you get all points with distance L to this surface, which is again a surface.
 
  • #4
As for part ii, you would get all points with distance m|r| to the plane normal to u and through the origin.

Let's start with m=0. What is it?
Now m=1: can you say in words which points you get?
 
  • #5
Since m is a fixed scalar, I don't think it changes. However, the modulus of r is increasing with distance from the origin. So do you get a kind of 3d parabola/ bowl shaped surface??
 
  • #6
Lucy Yeats said:
Since m is a fixed scalar, I don't think it changes. However, the modulus of r is increasing with distance from the origin. So do you get a kind of 3d parabola/ bowl shaped surface??

Ah, you're ahead of me (but no, it is not a bowl :wink:).

With m=0, you'd get r.u=0 which is a plane through the origin.

With m=1, you'd get all points r with a distance to the plane that is equal to the distance of r to the origin.
Which points would that be?
 
  • #7
All the points halfway between the origin and the surface?
 
  • #8
Lucy Yeats said:
All the points halfway between the origin and the surface?

No. The surface we're talking about contains the origin.
So there's no such thing as halfway.

Perhaps you can make a drawing in 2 dimensions.
Instead of a plane we'll have a line, but the principle remains the same.
Let u=(0,1).
What is the "plane" in this example?
Can you find a point that has an equal distance to the origin as to the line representing the "plane"?
 
  • #9
How do you get r˙uusing the length of the vectors and the angle they enclose? Try to use this formula.

ehild
 

Related to Identify these surfaces- quick vector question

1. What is the purpose of identifying surfaces in a quick vector question?

The purpose of identifying surfaces in a quick vector question is to understand the direction and magnitude of the vector in relation to the surface it is acting upon. This information is important in many scientific fields, such as physics and engineering, as it allows for accurate calculations and predictions.

2. How do you identify a surface in a vector question?

In order to identify a surface in a vector question, you need to determine the normal vector of the surface. This can be done by looking at the orientation of the surface and identifying the direction perpendicular to it. Once the normal vector is determined, it can be used to analyze the vector in question.

3. Can surfaces have multiple vectors acting upon them?

Yes, surfaces can have multiple vectors acting upon them. In this case, each vector would have its own normal vector and would need to be analyzed separately. It is important to consider all vectors acting upon a surface in order to accurately determine its overall effect.

4. Why is it important to consider surfaces in vector questions?

Considering surfaces in vector questions is important because it allows for a more accurate analysis of the situation. The surface can affect the direction and magnitude of the vector, and therefore, ignoring it can lead to incorrect calculations and predictions. In some cases, the surface itself may be the main focus of the question.

5. Are there any tips for quickly identifying surfaces in vector questions?

One helpful tip for quickly identifying surfaces in vector questions is to look for any visual cues, such as arrows or lines, that may indicate the orientation of the surface. Additionally, understanding the context of the question and the purpose of the vector can also help in identifying the surface it is acting upon.

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