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
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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
 

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