Exercise · ZW-PARAMETRIC-SURFACE-010 · ★★★ Advanced
On the cone \(\mathbf r(u,v)=\langle v\cos u,\,v\sin u,\,v\rangle\), at what value of \(v\) does \(\mathbf r_u\times\mathbf r_v\) vanish?
Choose one option.
Show answer
- A. \(v=0\), the vertex
Explanation
\(\mathbf r_u\times\mathbf r_v=\langle v\cos u,\,v\sin u,\,-v\rangle\), which is \(\mathbf 0\) only when \(v=0\): there the surface has no tangent plane.
Review the concept: Parametric surface
A parametric surface is the set of points in space described by a vector function \(\mathbf r\) of two parameters \(u\) and \(v\) as they range over a region \(D\) of the plane.
\[\mathbf r(u,v)=\langle x(u,v),\,y(u,v),\,z(u,v)\rangle,\quad (u,v)\in D\]
More practice problems: parametric surface
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- ZW-PARAMETRIC-SURFACE-004 A normal vector to mathbf r(u,v)=langle u, v, u^2+v^2rangle at (u,v)=(1,1) is:
- ZW-PARAMETRIC-SURFACE-005 For the cylinder mathbf r(u,v)=langlecos u, sin u, vrangle, the vector mathbf r_u×mathbf r_v is:
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