Exercises
- 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?
- Which of the points \((2,-1,7)\) or \((4,5,-3)\) or \((5,5,2)\) is closest to the origin? Which of those points is closest to the \(yz\)-plane?
- What is the equation of a sphere with center \((3,-1,6)\) and that passes through the point \((5,2,3)?\)
- Show that \(x^2+y^2+z^2+4x-6y+2z+6=0\) is the equation of a sphere, and find its center and radius.
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Each of the following statements regarding lines and planes and spheres in three-dimensional space is false. For each false statement determine all of possibilities that it does’t account for.
- A line must intersect a plane at a single point.
- A line must intersect a sphere at two points.
- Any two lines must intersect, and their intersection must be a point.
- Any two planes must intersect, and their intersection must be a line.
- If two lines are both perpendicular to a given line, then they’re parallel.
- If two lines are both parallel to a given plane, then they’re parallel.
- 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 \((1,2,4)\)?
- At what points does the plane \(3x-7y+z = 5\) intersect the coordinate axes?
- 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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Identify the surface in \(\mathbf{R}^3\) that corresponds to each of the following equations, by sketching a picture.
\(\displaystyle x=0\)\(\displaystyle z=3\)\(\displaystyle y=5\)\(\displaystyle y=x\)\(\displaystyle x = z\)\(\displaystyle y=x-1\)\(\displaystyle x+y = 3\)\(\displaystyle x+y+z = -2\)\(\displaystyle z=x^2\)\(\displaystyle x^2+y^2=1\)\(\displaystyle y^2+z^2=1\)\(\displaystyle x^2+y^2 = 4\)\(\displaystyle x^2+z^2 = 4\)\(\displaystyle x^2+4y^2 = 4\)\(\displaystyle x^2-y^2 = 4\)\(\displaystyle x^2+y^2+z^2 = 4\)\(\displaystyle x^2+y^2-z^2 = 4\)\(\displaystyle x^2+y^2-z^2 = 0\)\(\displaystyle x^2+y^2+z = 0\)\(\displaystyle x^2-y^2-z = 0\)\(\displaystyle x^2+\frac{y^2}{9}+\frac{z^2}{4}=1\)\(\displaystyle z=4x^2+y^2 \)\(\displaystyle z=y^2-x^2 \)\(\displaystyle \frac{x^2}{4}+y^2-\frac{z^2}{4}=1 \)\(\displaystyle \frac{x^2}{4}-y^2-\frac{z^2}{4}=1 \) -
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.
\(\rho = 17\)\(\theta = 1\)\(\phi = \frac{\pi}{5}\)\(\rho = \theta\)\(\rho = \phi\)
Problems & Challenges
- Convince yourself that the three-dimensional distance formula follow from applying the two-dimensional distance formula twice.
- 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?
- Consider the line that passes through the points \((2,4,-3)\) and \((3,-1,1),\) And consider the line that passes through the points \((-1,-3,4)\) and \((1,1,\zeta)\) for some variable \(\zeta.\) Does there exist a value of \(\zeta\) such that the lines are parallel? If so, what is it? If not, could you change the \(y\)-coordinate of that point so something else for which a value of \(\zeta\) making the lines parallel does exist?
- 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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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 \(\varphi\) is its angular distance
north-south from the earth’s equator,
and a location’s longitude \(\lambda\) is its angular distance
east-west from the prime meridian,
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,
which is about 39.07°N by 108.56°W in decimal.
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
(normal to the earth’s rotation),
the positive \(x\)-axis passes through the point
with GPS coordinates \(\bigl(\varphi, \lambda\bigr) = (0,0)\) on the earths surface
and the \(y\)-axis is oriented in accordance with the right-hand rule,
and note that the radius of the earth is approximately 3960 miles.
- 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?
- What is the city on the earth nearest to where the positive \(y\)-axis crosses its surface?
- 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 84.51°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 125.7475°E. How far apart along the surface of the earth are Grand Junction and Pyongyang?
- The approximate GPS coordinates of the city of Denver, CO are 39.74°N by 104.985°W. The approximate GPS coordinates of the city of Salt Lake City, UT are 40.76°N by 111.89°W. Along the surface of the earth how far are these cities from Grand Junction? Which one is closer? If a tunnel were bored straight through the earth to the nearer city, how much shorter would that tunnel be than the distance over the earth’s surface?
- 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.