91Ó°ÊÓ

Sketch the graph of the given cylindrical or spherical equation. $$ r=3 \cos \theta $$

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 12,12\rangle, \mathbf{v}=\langle-2,2\rangle $$

Find the symmetric equations of the line of intersection of the given pair of planes. \(x+4 y-2 z=13,2 x-y-2 z=5\)

find the unit tangent vector \(\mathbf{T}(t)\) and the curvature \(\kappa(t)\) at the point where \(t=t_{1} .\) For calculating \(\kappa\), we suggest using Theorem \(A\), as in Example \(5 .\) $$ x(t)=1-t^{2}, y(t)=1-t^{3} ; t_{1}=1 $$

Write the equation of the sphere with the given center and radius. (a) \((1,2,3) ; 5\) (b) \((-2,-3,-6) ; \sqrt{5}\) (c) \((\pi, e, \sqrt{2}) ; \sqrt{\pi}\)

Show that the vectors \(\langle 6,3\rangle\) and \(\langle-1,2\rangle\) are orthogonal.

Sketch the graph of the given cylindrical or spherical equation. $$ r=2 \sin 2 \theta $$

Show that the vectors \(\mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle 1,-1,0\rangle\), and \(\mathbf{c}=\langle-1,-1,2\rangle\) are mutually orthogonal, that is, each pair of vectors is orthogonal.

Find the symmetric equations of the line of intersection of the given pair of planes. \(x-3 y+z=-1,6 x-5 y+4 z=9\)

Find the equation of the plane through the given points. $$ (1,1,2),(0,0,1), \text { and }(-2,-3,0) $$