The Dot Product, Cross Product, and Vector Geometry

Trivium

Given two vectors, calculate the measure of the angle between them.

Given two vectors, calculate the vector that results from projecting one onto the other.

Given two intersecting planes in three-dimensional space, calculate the angle at which they intersect.

Given a plane and a point not on that plane, calculate the shortest distance from the point to the plane.

Given a line and a point not on that line, calculate the shortest distance from the point to the line.

Given two parallel planes in three-dimensional space, calculate the shortest distance between those planes.

Given two non-intersecting lines in three-dimensional space, calculate the shortest distance between those lines.

Given three non-colinear points in space, calculate the area of the triangle with those points as vertices.

Determine the unique plane that passes through any three non-colinear points in space.

Exercises

  1. Remember to replace h4 with span class="h5".

  2. 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 three points \((1,2,3)\) and \((6,4,5)\) and \((8,9,7)\) in \(\mathbf{R}^3.\)

    1. What is the area of the triangle with these points as vertices?
    2. Consider the three triangles that would result from projecting this triangle onto the coordinate planes. Which of these triangles has the largest area?
    3. There are three parallelograms in \(\mathbf{R}^3\) that have these three points as vertices. For each one, determine the coordinates of the fourth vertex. Why must all three of these parallelograms have the same area?
    4. What is the equation of the unique plane that contains these three points? Express the plane in the form \(Ax+By+Cz = D\) for whole numbers \(A\) and \(B\) and \(C\) and \(D.\)
    5. These three points determine a unique circle in \(\mathbf{R}^3.\) What is the area of this circle?
    6. Find an example of a point \(p\) in \(\mathbf{R}^3\) such that the tetrahedron formed by those three points and \(p\) has a volume of exactly one.
    7. Describe the locus of all points \(p\) in \(\mathbf{R}^3\) such that the tetrahedron formed by those three points and \(p\) has a volume of exactly one.
  2. Given two parallel planes \( Ax+By+Cz = D_1\) and \( Ax+By+Cz = D_2,\) find a formula for the shortest distance between them.
  3. Any two diagonals of a cube intersect in the center of the cube. What is the acute angle at which two diagonals of a cube intersect?
  4. These are some equations that are usually referred to as “properties of dot products” or “properties of cross products.”
    \(\bm{v}\cdot\bm{v} = |\bm{v}|^2\)
    \(\bm{u}\cdot(\bm{v}+\bm{w}) = \bm{u}\cdot\bm{v}+\bm{u}\cdot\bm{w}\)
    \((c\bm{u})\cdot\bm{v} = c(\bm{u}\cdot\bm{v}) = \bm{u}\cdot(c\bm{v})\)
    \((c\bm{u})\times\bm{v} = c(\bm{u}\times\bm{v}) = \bm{u}\times(c\bm{v})\)
    \(\bm{u}\times(\bm{v}+\bm{w}) = \bm{u}\times\bm{v}+\bm{u}\times\bm{w}\)
    \((\bm{v}+\bm{w})\times\bm{u} = \bm{v}\times\bm{u}+\bm{w}\times\bm{u}\)
    \(\bm{u}\cdot(\bm{v}\times\bm{w}) = (\bm{u}\times\bm{v})\cdot\bm{w} \)
    \(\bm{u}\times(\bm{v}\times\bm{w}) = (\bm{u}\cdot\bm{w})\bm{v} - (\bm{u}\cdot\bm{v})\bm{w} \)
    Can you come up with geometric intuition on why each of these properties holds true? Verifying these properties algebraically is not strictly necessary, but might be helpful to build the intuition.

  5. James Stewart

    A plane is capable of flying at a speed of 180 km/h in still air. The pilot takes off from an airfield and heads due north according to the plane’s compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually traveled 80 km at an angle 5° east of north.

    1. What is the wind velocity?
    2. In what direction should the pilot have headed to reach the intended destination?

  6. James Stewart

    Suppose the three coordinate planes are all mirrored and a light ray given by the vector \(\bm{a}= \langle a_1, a_2, a_3 \rangle\) first strikes the \(xz\)-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by \(\bm{b} = \langle a_1, -a_2, a_3\rangle.\) Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray.
    figure pending