91Ó°ÊÓ

Find the equation of the sphere that is tangent to the three coordinate planes if its radius is 6 and its center is in the first octant.

As you may have guessed, there is a simple formula for expressing great-circle distance directly in terms of longitude and latitude. Let \(\left(\alpha_{1}, \beta_{1}\right)\) and \(\left(\alpha_{2}, \beta_{2}\right)\) be the longitude- latitude coordinates of two points on the surface of the earth, where we interpret \(\mathrm{N}\) and \(\mathrm{E}\) as positive and \(\mathrm{S}\) and \(\mathrm{W}\) as negative. Show that the great-circle distance between these points is \(3960 \gamma\) miles, where \(0 \leq \gamma \leq \pi\) and $$ \cos \gamma=\cos \left(\alpha_{1}-\alpha_{2}\right) \cos \beta_{1} \cos \beta_{2}+\sin \beta_{1} \sin \beta_{2} $$

If \(\alpha=46^{\circ}\) and \(\beta=108^{\circ}\) are direction angles for a vector \(\mathbf{u}\), find two possible values for the third angle.

\mathbf{F}(t)=\mathbf{f}(u(t)) .\( Find \)\mathbf{F}^{\prime}(t)\( in terms of \)t$ $$ \text { } \mathbf{f}(u)=u^{2} \mathbf{i}+\sin ^{2} u \mathbf{j} \text { and } u(t)=\tan t $$

Find the equation of the sphere with center \((1,1,4)\) that is tangent to the plane \(x+y=12\).

Find two perpendicular vectors \(\mathbf{u}\) and \(\mathbf{v}\) such that each is also perpendicular to \(\mathbf{w}=\langle-4,2,5\rangle\).

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=3 t \mathbf{i}+3 t^{2} \mathbf{j} ; t_{1}=\frac{1}{3} $$

Evaluate the integrals $$ \int_{0}^{1}\left(e^{\prime} \mathbf{i}+e^{-t \mathbf{j}}\right) d t $$

Describe the graph in three-space of each equation. (a) \(z=2\) (b) \(x=y\) (c) \(x y=0\) (d) \(x y z=0\) (e) \(x^{2}+y^{2}=4\) (f) \(z=\sqrt{9-x^{2}-y^{2}}\)

Find the vector emanating from the origin whose terminal point is the midpoint of the segment joining \((3,2,-1)\) and \((5,-7,2)\).