Vector Trigonometry

Trivium

  1. Given two points in two- or three-dimensional space:
    1. Calculate the distance between those points, and their midpoint.
    2. Determine a parameterization of the line that contains those points.
  2. Given two vectors in two- or three-dimensional space:
    1. Calculate and sketch any linear combination of those vectors.
    2. Calculate the measure of the angle formed between those vectors.
    3. Calculate the vector resulting from projecting one vector onto the other.
    4. Calculate a vector that is orthogonal to both of of those vectors.
  3. Determine an equation for the plane that contains thee non-colinear points.
  4. Determine a parameterization of the line along which two planes intersect.
  5. Determine the point at which a plane is intersected by a line not in that plane.
  6. Compute the shortest distance to a plane from a point not on that plane.
  7. Compute the shortest distance to a line from a point not on that line.
  8. Compute the shortest distance between two skew lines.

Problems

  1. Suppose an airplane plans to travel 456 mph with a heading of N30°W. How do you describe this velocity/trajectory as a vector? I.e. what are the horizontal and vertical components of the plane’s velocity?
  2. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph. What is the plane’s heading as an angle? What is the plane’s speed?
  3. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but there is a steady 40 mph wind blowing from the southeast. What is the course of the plane as a vector? What is the actual speed of the plane?
  4. Suppose an airplane is flying at a constant speed of 430 mph and has an intended course due east-northeast. However there is a steady 40 mph wind blowing from the north. What should the plane’s heading be to stay on course?
  5. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but due to the effects of a heavy wind, actually has a course of \(\langle 255, 377 \rangle\) mph. What is the velocity vector of the wind? What is the speed of the wind?
  6. Suppose you must swim across a straight river that is 20 ft wide, and you want to end up at the spot directly across the river from where you are now. The river has a constant current speed of 3 ft/s, but luckily you can swim at about 5 ft/s. In what direction do you need to swim to counteract the current to ensure you land directly across the river?
  7. What is the area of the triangle in the plane with vertices located at the coordinates \((1,1)\) and \((3,-3)\) and \((2,7)?\)
  8. What is the area of the triangle in space with vertices located at the coordinates \((1,1,1)\) and \((3,-3,8)\) and \((2,7,0)?\)
  9. Consider the points \((1,1,1)\) and \((3,-3,8).\) Determine a point on the \(z\)-axis that is equidistant to those points.
  10. 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?
  11. What is an equation of the line in three-dimensional space that passes through the point \((5,6,7),\) intersects the \(z\)-axis, and makes an angle of \(39°\) with the \(xy\)-plane?
  12. What is an equation of the plane that intersects the \(xy\)-plane along the line \(y=2x-1\) in the \(xy\)-plane and is inclined at an angle of \(39°\) from the \(xy\)-plane?
  13. Any two diagonals of a cube intersect in the center of the cube. What is the acute angle at which two of those diagonals intersect?
  14. What is the measure of the acute angle formed at the intersection of the planes \(-3x + 5y + z= 35\) and \(7x - 2y + z = 6?\) In general, given two planes \(A_1x + B_1y + C_1z = D_1\) and \(A_2x + B_2y + C_2z = D_2,\) devise a general formula for the measure of the angle at which they intersect
  15. Devise a formula for the distance from a point \(P\) to a line \({\bm{v}t\!+\!Q.}\)
  16. Devise a formula for the distance from a point \(P\) to a plane \({Ax\!+\!By\!+\!Cz\!=\!D.}\)
  17. Devise a formula for the distance between the lines \({\bm{u}t\!+\!P}\) and \({\bm{v}t\!+\!Q.}\)

  18. TK A tetrahedron is a three-dimensional shape having four triangular face, six edges, and four vertices. It is the three-dimensional analogue of a triangle, and may be visualized a triangular-based pyramid. The plane \(3(x-1) + 2(y-5) + (z-4) = 0,\) along with the three coordinate planes, bounds a tetrahedron in space. What is the volume of this tetrahedron?

    In general, for \(A, B, C \neq 0,\) what’s a formula for the volume of the tetrahedron bound by the coordinate planes and the plane \(Ax + By + Cz = D.\)

  19. An implication of Heron’s formula is that the area of a triangle is uniquely determined by the lengths of its sides. Is either analogous statement for a tetrahedron true? Given the six side-lengths of a tetrahedron is the volume uniquely determined? Or given the areas of the four faces of a tetrahedron is the volume uniquely determined?

  20. Given three numbers \(A\) and \(B\) and \(C\) such that \(A \lt B \lt C,\) there is only a triangle with those numbers as side-lengths if the numbers \(A\) and \(B\) and \(C\) satisfy the triangle inequality \(B-A \lt C \lt B+A.\)

    What is the analogous statement for a tetrahedron in three-dimensional space? Given six numbers \(A, B, C, D, E, F\) that are the side-lengths of a tetrahedron, what requirements must they satisfy? Or given four numbers \(M, N, O, P\) that are the areas of the triangular faces of a tetrahedron, what requirements must they satisfy?

  21. Mr Goat is tethered to a stake in the ground at the concave corner of an “L”-shaped shed that is six feet tall. an image of how Mr Goat is tethered to the shed. Idk how to describe it accurately, so I'll work on making this an SVG file instead some day. The Genie of Goats happens to wander by and offer Mr Goat a wish, at which Mr Goat immediately wishes for the ability to fly. At a snap of his goat-fingers, the Genie of Goats grants his wish and disappears. Mr Goat can now fly! However he’s still tethered o the stake. Considering the length of Mr Goat’s tether and the dimensions of the shed indicated in the image, What is the volume of the region in which Mr Goat can fly around? Challenge: Suppose instead of that inner corner, Mr Goat were tethered to the upper-right corner indicated with a small square. Now what is the volume of the region in which Mr Goat can fly around?
  22. A right-angled halway corner as described in the prompt Suppose you have to transport a bunch of long, heavy, cast iron pipes one-by-one down a 7′ wide 8′ tall hallway, around a right-angled corner, and into a 5′ wide 8′ tall tall hallway. The pipes are awkward to carry around the corner but you can tilt them and carry them at an angle to make the task easier. You realize that getting longer pipes around the corner is going to be tough, and will be downright impossible if the pipe is too long. What’s the longest length of a pipe that you could possibly swivel around that corner?