91Ó°ÊÓ

In Exercises 29–34, find an equation of the line containing the given pair of points. Express your answer (a) as a vector parametrization, (b) in terms of parametric equations, and (c) in symmetric form.

P(1, 6), Q(4, 5)

In Exercises 30–35 compute the indicated quantities when $u=(−3,1,−4),v=(2,0,5),\mathrm{and}w=(1,3,13)$

role="math" localid="1649436488889" $(u×v)·w\mathrm{and}u·(v×w)$

Find the equation of a sphere containing the point (1, 4, 7) and whose center is (-2, 3, 5).

Find the norm of the vector.$v=\left(\frac{1}{2},\frac{1}{3},\frac{1}{4}\right)$

Show that the lines determined by

${\mathbf{r}}_1\left(t\right)=⟨3-t,4-4t,-3+4tâŸ\copyright$and ${\mathbf{r}}_2\left(t\right)=⟨5-t,-6+2t,-2+tâŸ\copyright$

intersect, and then find an equation of the plane containing both lines.

Find the volume of the parallelepiped determined by \(u=<2,4,-1>\), \(v=<0,-3,2>\), and \(w=<-1,1,5>\).

In Exercises 29–34, find an equation of the line containing the given pair of points. Express your answer (a) as a vector parametrization, (b) in terms of parametric equations, and (c) in symmetric form.

P(3, −1, 2, 6), Q(1, 4, 5, −2)

In Exercises 30–35 compute the indicated quantities when $u=(−3,1,−4),v=(2,0,5),\mathrm{and}w=(1,3,13)$

Find the area of the parallelogram determined by the vectors u and v.

Find the equation of a sphere containing the point (3, 0, -1) and whose center is (2, -8, 0).

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems

Change$\mathrm{ÒÏ}=2$ to the rectangular and cylindrical systems