(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Identify the following surface. I'm not sure what the answer is, but I believe I am on the right track, appreciate if anyone can check this for me.

[tex]{\bf r} \cdot {\bf u} = m |{\bf r}| , -1 \le {\bf m} \le 1 [/tex]

2. Relevant equations

Nothing really.

3. The attempt at a solution

Split cases. For m=0, [tex]{\bf r} \cdot {\bf u} = {\bf 0} [/tex], the locus of which is the origin.

For m = 1, [tex]{\bf r} \cdot {\bf u} = |{\bf r}| [/tex]. The equation describes a plane (whose normal vector is u) distance [tex]|{\bf r}|[/tex] from the origin. For m = -1, the equation describes a plane an equal distance from the origin and in the opposite direction. The locus described is any vector whose projection on the plane is [tex]|{\bf r}|[/tex] from the origin, forming the locus of a circle.

I'm already about halfway there I think. As the values of m are continuous, the surface so generated is a cone with the axis parallel to u. But I also need to define the semi-parallel angle.

4. More attempt

[tex]{\bf \hat{r}} \cdot {\bf u} = m[/tex], and [tex]{\bf \hat r} = 1[/tex] Therefore, [tex]cos\theta = m[/tex]?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Surface defined by vector product

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