91Ó°ÊÓ

Find the given distances. The distance between the parallel planes \(2(x-1)+2(y+1)+(z-2)=0\) and $$ 2(x-3)+2(y-1)+(z-3)=0 $$

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Write \(\vec{u}\) as the sum of two vectors, one of which is parallel to \(\vec{v}\) and one of which is perpendicular to \(\vec{v}\). Note: these are the same pairs of vectors as found in Exercises 21-26. \(\vec{u}=\langle-3,2\rangle, \vec{v}=\langle 2,3\rangle\)

Sketch the quadric surface. \(\frac{x^{2}}{9}-y^{2}+\frac{z^{2}}{25}=1\)

In Exercises 31-32, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.) Vertices: (0,0),(1,2),(3,0) and (4,3) .

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Write \(\vec{u}\) as the sum of two vectors, one of which is parallel to \(\vec{v}\) and one of which is perpendicular to \(\vec{v}\). Note: these are the same pairs of vectors as found in Exercises 21-26. \(\vec{u}=\langle 3,-1,2\rangle, \vec{v}=\langle 2,2,1\rangle\)

Find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.) Vertices: (0,0,0),(2,1,1),(-1,2,-8) and (1,-1,5) .

Sketch the quadric surface. \(4 x^{2}+2 y^{2}+z^{2}=4\)

In Exercises 33-34, find the volume of the parallelepiped defined by the given vectors. \(\vec{u}=\langle 1,1,1\rangle, \quad \vec{v}=\langle 1,2,3\rangle, \quad \vec{w}=\langle 1,0,1\rangle\)

Find the volume of the parallelepiped defined by the given vectors. \(\vec{u}=\langle-1,2,1\rangle, \quad \vec{v}=\langle 2,2,1\rangle, \quad \vec{w}=\langle 3,1,3\rangle\)

A 10lb box sits on a \(15 \mathrm{ft}\) ramp that makes a \(30^{\circ}\) angle with the horizontal. How much force is required to keep the box from sliding down the ramp?