Trivium
Determine the rectangular, cylindrical, or spherical coordinates of a point given the coordinates of the point in any one of those systems.
For two points in space, calculate the distance between those points and their midpoint, and determine a parameterization or vector equation of the line that contains those points.
Exercises
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Given these points expressed in rectangular coordinates,
write down their cylindrical coordinates.
Then plot each point, making sure the location of the point
makes sense in both rectangular and cylindrical coordinates.
\((-0.415, 0.909, 3)\)\((1.081, 1.683, -3)\)\((-2.912, -6.363, 4)\)\((-2.075, 4.545, 0)\)\((-0.989, 0.141, 314)\)
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Equipping space with a cylindrical coordinate system,
each of these equations describes a surface in space.
Determine what that surface must look like.
\(r = 42\)\(\theta = \frac{\pi}{4}\)\(z = -31\)\(r = z\)\(r = z^2\)\(r = \theta\)
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Given these points expressed in rectangular coordinates,
write down their spherical coordinates.
Then plot each point, making sure the location of the point
makes sense in both rectangular and spherical coordinates.
\((0.454, 0.707, 0.540)\)\((2.727, 0, -1.245)\)\((2.270, 3.536, 2.700)\)\((0, 0, 5)\)\((-1.535, -0.218, -10.879)\)
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Equipping space with a spherical coordinate system,
each of these equations describes a surface in space.
Determine what that surface must look like.
\(\rho = 17\)\(\theta = 1\)\(\phi = \frac{\pi}{5}\)\(\rho = \theta\)\(\rho = \phi\)
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Sketch the surface in \(\mathbf{R}^3\)
that corresponds to each of the following equations.
\(x=0\)\(z=3\)\(y=5\)\(y=x\)\(x = z\)\(y=x-1\)\(x+y = 3\)\(x+y+z = -2\)\(x^2+y^2 = 4\)\(x^2+z^2 = 4\)\(x^2+4y^2 = 4\)\(x^2-y^2 = 4\)\(x^2+y^2+z^2 = 4\)\( x^2+y^2-z^2 = 4\)\( x^2+y^2-z^2 = 0\)\( x^2+y^2+z = 0\)\( x^2-y^2-z = 0\)
- Convince yourself that the three-dimensional distance formula follow from applying the two-dimensional distance formula twice.
- What is the distance between the points \((2,-1,7)\) and \((1,-3,5)?\) What is the midpoint of the segment between those points?
- Consider a line segment with one endpoint at the coordinates \((4,1,2)\) and midpoint at the coordinates \((-3,4,5).\) What are the coordinates of the other endpoint?
- What is the equation of a sphere with center \((3,-1,6)\) and that passes through the point \((5,2,3)?\)
- Find parametric equations and symmetric equations of the line that passes through the points \((2,4,-3)\) and \((3,-1,1).\) At what point does this line intersect the \(xy\)-plane?
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Show that the lines with the following parametric equations
are skew lines — they do not intersect.
\((x,y,z) = \bigl(1+t, -2+3t, 4-t\bigr)\)\((x,y,z) = \bigl(2t, 3+t, -3+4t\bigr)\)
- What’s an equation for the plane that intersects the \(xy\)-plane along the line \(2x-3y=4\) and contains the point \((1,2,4)\)?
- Find the point at which the line with parametric equations \((x,y,z) = {\bigl(2+3t, -4t, 5+t\bigr)}\) intersects the plane \(4x+5y-2z=18.\)
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What surface in \(\mathbf{R}^3\)
is corresponds to each of the following equations?
\(\displaystyle z=x^2\)\(\displaystyle x^2+y^2=1\)\(\displaystyle y^2+z^2=1\)\(\displaystyle x^2+y^2+z^2=49\)\(\displaystyle x^2+\frac{y^2}{9}+\frac{z^2}{4}=1\)\(\displaystyle z=4x^2+y^2 \)\(\displaystyle z=y^2-x^2 \)\(\displaystyle \frac{x^2}{4}+y^2-\frac{z^2}{4}=1 \)\(\displaystyle \frac{x^2}{4}-y^2-\frac{z^2}{4}=1 \)
Problems & Challenges
- Consider the points \((1,1,1)\) and \((3,-3,8).\) Determine a point on the \(z\)-axis that is equidistant to those points.
- Describe the set of all points that are equidistant (the same distance) from the points \((1,2,3)\) and \((-6,5,4).\) Can you find an equation that corresponds to this set? I.e. can you find the equation that relates any \(x\) and \(y\) and \(z\) such that \((x,y,z)\) is equidistant from those points?
- Describe the set of all points that are twice the distance to \((1,2,3)\) as they are to \((-6,5,4).\) Can you find an equation that corresponds to this set?
- What is an equation for the plane that crosses the \(x\)-axis at \(3\) and crosses the \(y\)-axis at \(5\) and crosses the \(z\)-axis at \(11?\)
- The concept of the slope of a line still kinda makes sense in three-dimensional space as “rise over run” where the “rise” is the change in position of the line in the \(z\) direction and “run” is the change in the position of the line parallel to the \(xy\)-plane. What is the slope of the line described parametrically as \[ (x,y,z) = \bigl(1+2t, 3-3t, 7+5t\bigl)\,? \]
- We’ve developed formulas for converting between rectangular and cylindrical coordinates, and between rectangular and spherical coordinates, Now write down formulas that convert directly between cylindrical and spherical coordinates.
- Are there any triples \((a,b,c)\) that represent the same point in space whether interpreted as a location in rectangular or cylindrical coordinates? What about in rectangular and spherical coordinates?
- The earth is approximately spherical, with a radius of 3960 miles. Classically, a nautical mile was defined to be the distance along the earth’s surface subtended by a central angle of 1′, i.e. one-sixtieth of a degree. How many miles are in a nautical mile?
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The earth is approximately spherical, with a radius of 3960 miles.
Our convention when defining spherical coordinates
does not match up with the convention for defining
Global Positioning System (GPS) coordinates on earth.
A location’s latitude is its angular distance
north-south from the earth’s equator,
and a location’s longitude is its angular distance
east-west from the prime meridian,
which is the meridian passing through the Royal Observatory in Greenwich, London.
For example, the city of Grand Junction CO
has a latitude of 39°05′16″N and a longitude of 108°34′05″W.
Suppose the earth is oriented in space such that
the center of the earth is at the origin,
the positive \(z\)-axis is aligned with the north pole,
the positive \(x\)-axis passes through the point
with GPS coordinates \((\phi, \lambda) = (0,0)\) on the earths surface
and the \(y\)-axis is oriented in accordance with the right-hand rule.
- What must be true of a triple \((x,y,z)\) in rectangular coordinates to guarantee that point is on the surface of the earth? (instead of within the earth, or floating out in space, for instance)
- Given a triple \((x,y,z)\) in rectangular coordinates that corresponds to a point on earth, what is the latitude \(\varphi\) and longitude \(\lambda\) of that point?
- Given the GPS coordinates \(\varphi\) and \(\lambda\) of a point in space on the earth’s surface, what are it’s rectangular (absolute) coordinates?
- Where on earth’s surface does the positive \(y\)-axis cross?
- How far along the surface of the earth would a person have to travel south from Grand Junction before they reached the equator?
- The city of Cincinnati, Ohio has approximately the same latitude as Grand Junction but a longitude of 84°30′45″W. How far apart along the surface of the earth are Grand Junction and Cincinnati?
- The city of Pyongyang, North Korea has approximately the same latitude as Grand Junction but a longitude of 125°44′51″E. How far apart along the surface of the earth are Grand Junction and Pyongyang?
- The approximate GPS coordinates of the city of Anchorage, Alaska are 61°13′3″N and 149°51′47″W. How far apart along the surface of the earth are Grand Junction and Anchorage? If you were to bore a straight tunnel through the earth between Grand Junction and Anchorage, how long would that tunnel have to be?
- The earth is approximately a sphere, but only approximately. Due to the earth’s rotation and centrifugal force, it’s “bulging” at the equator and would be more accurately described as an ellipsoid (or oblate spheroid). The current World Geodesic System standard, WGS-84, establishes the distance from the earth’s center to the equator as 3963.191 miles whereas the distance from the earth’s center to either pole is 3949.903 miles. Write an equation that describes a surface in three-dimensional space that models the ellipsoidal surface of this earth where its center is at the origin \((0,0,0)\) and the north pole lies on the \(z\)-axis.