Chapter 12

Convert the following points in rectangular coordinates to cylindrical and spherical coordinates: (a) (1,1,1) (b) (7,-7,5) (c) \((\cos (1), \sin (1), 1)\) (d) \((0,0,-\pi)\)

Find an equation of the plane containing (6,2,1) and perpendicular to \(\langle 1,1,1\rangle .\)

Find the cross product of \langle 1,1,1\rangle and \(\langle 1,2,3\rangle .\)

Find \(\langle 1,1,1\)\rangle\(\cdot\langle 2,-3,4\rangle\)

Draw the vector \langle 3,-1\rangle with its tail at the origin.

Sketch the location of the points \((1,1,0),(2,3,-1),\) and (-1,2,3) on a single set of axes.

Find an equation for the sphere \(x^{2}+y^{2}+z^{2}=4\) in cylindrical coordinates.

Find an equation of the plane containing (-1,2,-3) and perpendicular to \(\langle 4,5,-1\rangle .\)

Find the cross product of \langle 1,0,2\rangle and \(\langle-1,-2,4\rangle .\)

Find \(\langle 1,2,0\rangle \cdot\langle 0,0,57\rangle .\)

Draw the vector \(\langle 3,-1,2\rangle\) with its tail at the origin.

Describe geometrically the set of points \((x, y, z)\) that satisfy \(z=4\)

Find an equation for the yz-plane in cylindrical coordinates.

Find an equation of the plane containing (1,2,-3),(0,1,-2) and (1,2,-2) .

Find the cross product of \langle-2,1,3\rangle and \langle 5,2,-1\rangle .

Find \(\langle 3,2,1\rangle \cdot\langle 0,1,0\rangle .\)

Let \(v\) be the vector with tail at the origin and head at (1,2)\(;\) let \(w\) be the vector with tail at the origin and head at \((3,1) .\) Draw \(v\) and \(w\) and a vector \(u\) with tail at (1,2) and head at \((3,1) .\) Draw \(\boldsymbol{u}\) with its tail at the origin.

Describe geometrically the set of points \((x, y, z)\) that satisfy \(y=-3 .\)

Find an equation equivalent to \(x^{2}+y^{2}+2 z^{2}+2 z-5=0\) in cylindrical coordinates.

Find an equation of the plane containing (1,0,0),(4,2,0) and \((3,2,1) .\)

Find the cross product of \langle 1,0,0\rangle and \(\langle 0,0,1\rangle .\)

Find \langle-1,-2,5\rangle\(\cdot\langle 1,0,-1\rangle\)

Let \(v\) be the vector with tail at the origin and head at (-1,2)\(;\) let \(w\) be the vector with tail at the origin and head at \((3,3) .\) Draw \(v\) and \(w\) and a vector \(u\) with tail at (-1,2) and head at (3,3) . Draw \(\boldsymbol{u}\) with its tail at the origin.

Describe geometrically the set of points \((x, y, z)\) that satisfy \(x+y=2\).

Suppose the curve \(z=e^{-x^{2}}\) in the xz-plane is rotated around the z-axis. Find an equation for the resulting surface in cylindrical coordinates.

Find an equation of the plane containing (1,0,0) and the line \(\langle 1,0,2\rangle+t\langle 3,2,1\rangle .\)

Two vectors \(\boldsymbol{u}\) and \(\boldsymbol{v}\) are separated by an angle of \(\pi / 6,\) and \(|\boldsymbol{u}|=2\) and \(|\boldsymbol{v}|=3\). Find \(|\boldsymbol{u} \times \boldsymbol{v}|\)

Find \(\langle 3,4,6\)\rangle$$\cdot\langle \(2,3,4\)\rangle$

Let \(v\) be the vector with tail at the origin and head at (5,2) ; let \(w\) be the vector with tail at the origin and head at \((1,5) .\) Draw \(v\) and \(w\) and a vector \(u\) with tail at (5,2) and head at \((1,5) .\) Draw \(\boldsymbol{u}\) with its tail at the origin.

The equation \(x+y+z=1\) describes some collection of points in \(\mathbb{R}^{3} .\) Describe and sketch the points that satisfy \(x+y+z=1\) and are in the xy- plane, in the xz-plane, and in the yz-plane.

Suppose the curve \(z=x\) in the \(x z\) -plane is rotated around the z-axis. Find an equation for the resulting surface in cylindrical coordinates.

Find an equation of the plane containing the line of intersection of \(x+y+z=1\) and \(x-y+2 z=2,\) and perpendicular to the xy-plane.

Two vectors \(\boldsymbol{u}\) and \(\boldsymbol{v}\) are separated by an angle of \(\pi / 4,\) and \(|\boldsymbol{u}|=3\) and \(|\boldsymbol{v}|=7\). Find \(|\boldsymbol{u} \times \boldsymbol{v}| .\)

Exercise 12.3.6 Find the cosine of the angle between \langle 1,2,3\rangle and \langle 1,1,1\rangle\(;\) use a calculator if necessary to find the angle.

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v}-\boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 1,3\rangle\) and \(\boldsymbol{w}=\langle-1,-5\rangle .\)

Find the lengths of the sides of the triangle with vertices \((1,0,1),(2,2,-1),\) and \((-3,2,-2) .\)

Find an equation for the plane \(y=0\) in spherical coordinates.

Find the area of the parallelogram with vertices \((0,0),(1,2),(3,7),\) and \((2,5) .\)

Find the cosine of the angle between \langle-1,-2,-3\rangle and \langle 5,0,2\rangle\(;\) use a calculator if necessary to find the angle.

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v} \boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 1,2,3\rangle\) and \(\boldsymbol{w}=\langle-1,2,-3\rangle .\)

Find the lengths of the sides of the triangle with vertices \((2,2,3),(8,6,5),\) and \((-1,0,2) .\) Why do the results tell you that this isn't really a triangle?

Find an equation for the plane \(z=1\) in spherical coordinates.

Find an equation of the line through (1,0,3) and perpendicular to the plane \(x+2 y-z=\) \(1 .\)

Find and explain the value of \((\boldsymbol{i} \times \boldsymbol{j}) \times \boldsymbol{k}\) and \((\boldsymbol{i}+\boldsymbol{j}) \times(\boldsymbol{i}-\boldsymbol{j}) .\)

Find the cosine of the angle between \langle 47,100,0\rangle and \langle 0,0,5\rangle\(;\) use a calculator if necessary to find the angle.

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v}-\boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 1,0,1\rangle\) and \(\boldsymbol{w}=\langle-1,-2,2\rangle .\)

Find an equation of the sphere with center at (1,1,1) and radius 2 .

Find an equation for the sphere with radius 1 and center at (0,1,0) in spherical coordinates.

Find an equation of the line through the origin and perpendicular to the plane \(x+y-z=\) 2.

Prove that for all vectors \(\boldsymbol{u}\) and \(\boldsymbol{v},(\boldsymbol{u} \times \boldsymbol{v}) \cdot \boldsymbol{v}=0 .\)