# Identify these surfaces- quick vector question

• Lucy Yeats
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.

## Homework Statement

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

## 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.

Is part i just an ordinary plane?

Hi Lucy Yeats!

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.

As for part ii, you would get all points with distance m|r| to the plane normal to u and through the origin.

Now m=1: can you say in words which points you get?

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??

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 ).

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?

All the points halfway between the origin and the surface?

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"?

How do you get r˙uusing the length of the vectors and the angle they enclose? Try to use this formula.

ehild