91Ó°ÊÓ

For Exercises \(1-4,\) determine if the given equation deseribes a sphere. If so, find its radius and center. $$ x^{2}+y^{2}+z^{2}-4 x-6 y-10 z+37=0 $$

For Exercises \(1-4,\) determine if the given equation deseribes a sphere. If so, find its radius and center. $$ x^{2}+y^{2}+z^{2}+2 x-2 y-8 z+19=0 $$

For Exercises \(1-4,\) calculate \(\mathbf{f}^{\prime}(t)\) and find the tangent line at \(\mathrm{f}(0)\). $$ \mathbf{f}(t)=\left(e^{t}+1, e^{2 t}+1, e^{t^{2}}+1\right) $$

For Exereises \(1-4,\) find the (a) cylindrical and (b) spherical coordinates of the point whose Cartesian coordinates are given. $$ (-5,5,6) $$

Write the line \(L\) through the point \(P\) and parallel to the vector \(\mathbf{v}\) in the following forms: (a) vector, (b) parametric, and (c) symmetric. \(P=(3,-1,2), \mathbf{v}=(2,8,1)\)

For Exercises \(1-4,\) calculate \(\mathbf{f}^{\prime}(t)\) and find the tangent line at \(\mathrm{f}(0)\). $$ \mathbf{f}(t)=(\cos 2 t, \sin 2 t, t) $$

For Exercises \(11-12,\) find the volume of the parallelepiped with adjacent sides \(\mathbf{u}, \mathbf{v}, \mathbf{w}\). $$ \mathbf{u}=(1,1,3), \mathbf{v}=(2,1,4), \mathbf{w}=(5,1,-2) $$

Write the normal form of the plane \(P\) containing the point \(Q\) and perpendicular to the vector \(\mathbf{n}\). $$ Q=(5,1,-2), \mathbf{n}=(4,-4,3) $$

Show that the hyperboloid of one sheet is a doubly ruled surface, i.e. each point on the surface is on two lines lying entirely on the surface. Hint: Write equation (1.35) as \(\frac{x^{2}}{a^{2}}-\frac{z^{2}}{e^{2}}=1-\frac{y^{2}}{b^{2}},\) factor each side. Recall that two planes intersect in a line)

Show that the distance \(d\) between the points \(P_{1}\) and \(P_{2}\) with cylindrical coordinates \(\left(r_{1}, \theta_{1}, z_{1}\right)\) and \(\left(r_{2}, \theta_{z}, z_{2}\right),\) respectively, is $$ d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)+\left(z_{2}-z_{1}\right)^{2}} $$