Lines and Planes in Three-Dimensional Space

Exercises

  1. What is an equation of the plane with normal vector \(\langle 2,3,-5\rangle\) that contains the point \((1,-2,3)?\)
  2. What is an equation of the plane that passes through the coordinates axes at \(x=4\) and \(y=-1\) and \(z=3?\)
  3. What is an equation of the plane containing the three points \((1,-2,3)\) and \((3,1,-4)\) and \((0,7,1)?\) What are the coordinates of the points where this plane intersects the coordinate axes?
  4. What is a parameterization of the line passing through \((1,-2,3)\) in the direction \(\langle 4,-3,2 \rangle?\) What are the coordinates of the points where this line crosses the coordinate planes?
  5. What is a parameterization of the line passing through the points \((1,-2,3)\) and \((9,-8,-7)?\) What are the coordinates of the points where this line crosses the coordinate planes?
  6. What is a parameterization of the line along which the planes \(x-2(y-3)+4(z+1)=0\) and \(3(x-1)+y-z=0\) intersect?
  7. What are the coordinates of the point where the plane \(x-2(y-3)+4(z+1)=0\) and the line \(\langle 2,-6,1\rangle t + (1,4,4)\) intersect?

Problems & Challenges

  1. 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?
  2. 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?
  3. 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?
  4. 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

  5. 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 \(Ax + By + Cz = D.\)

    An implication of Heron’s formula is that the area of a triangle is uniquely determined by the lengths of its sides. 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?

    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?