Three-Dimensional Coordinate Systems & Vectors

Trivium

Determine the rectangular, cylindrical, or spherical coordinates of a point given the coordinates of the point in any one of those systems.

Given two points in two- or three-dimensional space, Calculate the distance between those points, and their midpoint. Determine a parameterization of the line that contains those points.

Given two vectors, calculate and sketch any linear combination of those vectors,

[TK] Vectors?

Exercises

  1. [TK] Gather example from Stewarts.
  2. [TK] Lowkey just plot points and get oriented.
  3. [TK] See /exercises and gather *some* from Stewart.
  4. [TK] See /exercises/spherical-cylindrical.html
  5. Remember to replace h4 with span class="h5".

  6. TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK

    \(\displaystyle y = mx+b\)
    \(\displaystyle y = mx+b\)

Problems & Challenges

  1. Consider the points \((1,1,1)\) and \((3,-3,8).\) Determine a point on the \(z\)-axis that is equidistant to those points.
  2. 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?
  3. 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?
  4. 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?
  5. 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 \(11?\)
  6. Considering the plane given by the equation \( Ax+By+Cz = D,\) we know geometric interpretation for those coefficients \(A\) and \(B\) and \(C\) — they are the coordinates of a vector normal to the plane. But what about \(D?\) How can we geometrically interpret \(D?\)
  7. Are there any triples \((a,b,c)\) that represent the same point in space whether interpreted as a location in rectilinear or cylindrical coordinates? What about in rectilinear and spherical coordinates?

  8. 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 6,378.1370 km, whereas the distance from the earth’s center to either pole is 6,356.7523 km.

    1. TK

    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.

  9. The convention when defining spherical coordinates does not match up with our convention for defining GPS coordinates (longitude and latitude) on earth. Like how we have formulas to convert between rectangular coordinates and spherical coordinate, we should develop formulas to convert between rectangular coordinates and GPS coordinate. Look up how GPS coordinates describe a point on the earth. 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 \((0,0)\) 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?