The Dot Product, Cross Product, and Vector Geometry

Trivium

Given two vectors in space:

• Calculate and sketch any linear combination of those vectors.
• Calculate the measure of the angle formed between those vectors.
• Calculate the area of the triangle framed by those vectors.
• Calculate the vector resulting from projecting one vector onto the other.
• Calculate a vector that is orthogonal to both of of those vectors.

Determine an equation for the plane that contains three non-colinear points.

Determine a parameterization of the line along which two planes intersect.

Determine the point at which a plane is intersected by a line not in that plane.

Compute the shortest distance to a plane from a point not on that plane.

Compute the shortest distance to a line from a point not on that line.

Compute the shortest distance between two skew lines.

Exercises

1. If you start at the initial point \((3,7,1)\) and travel to the terminal point \((1,2,5),\) along what vector did you travel?
2. If you start at the initial point \((-2,7,6)\) and travel along a vector \(\langle -1,3,0 \rangle,\) what are the coordinates of the terminal point?
3. Sketch the vectors \(\bm{u} = \langle 3,-3,5 \rangle\) and \(\bm{v} = \langle 2,5,-4 \rangle\) in \(xyz\)-space, then explicitly calculate each of the following:

   \( \bm{u} + \bm{v} \)

   \( \bm{u} - \bm{v} \)

   \( |\bm{u}| \)

   \( \bm{\hat{u}} \)

   \( \bm{\hat{v}} \)

   \( \bm{u} - 2\bm{v} \)

   \( \bm{v} + \frac{1}{2}\bm{u} \)

   \( \bm{u} + 17\mathbf{j} \)

4. What are the values of the following dot products?

   \(\displaystyle \left\langle 2,4 \right\rangle\cdot\left\langle 3,-1 \right\rangle \)

   \(\displaystyle \left\langle -1,7,4 \right\rangle\cdot\left\langle 6,2,-1/2 \right\rangle \)

   \(\displaystyle \left(\mathbf{i}+2\mathbf{j}-3\mathbf{k}\right)\cdot\left(2\mathbf{j}-\mathbf{k}\right) \)

5. Suppose that \(|\bm{u}| = 4\) and \(|\bm{v}| = 6\) and the angle between them is \(\pi/3.\) What must the value of their dot product be?
6. What is the angle between the vectors \(\langle 2,2,1 \rangle\) and \(\langle 5,-3,2 \rangle?\)
7. For the vectors \(\bm{u} = \langle 1,1,2 \rangle\) and \(\bm{v} = \langle -2,3,1 \rangle,\) what is \(\operatorname{proj}_{\bm{v}}(\bm{u}),\) the projection of \(\bm{u}\) onto \(\bm{v}?\)
8. What are the values of the following cross products?

   \(\displaystyle \left\langle 1,3,4 \right\rangle\times\left\langle 2,7,-5 \right\rangle \)

   \(\displaystyle \left\langle -1,7,4 \right\rangle\times\left\langle 6,2,-1/2 \right\rangle \)

   \(\displaystyle \left(\mathbf{i}+2\mathbf{j}-3\mathbf{k}\right)\times\left(2\mathbf{j}-\mathbf{k}\right) \)

9. Consider the three points \((1,4,6)\) and \((-2,5,-1)\) and \((1,-1,1)\)
   1. Determine a vector perpendicular to the plane that contains them.
   2. What is the area of the triangle with those three points as vertices?
10. Show that the vectors \(\langle 1,4,-7 \rangle\) and \(\langle 2,-1,4 \rangle\) and \(\langle 0,-9,18 \rangle\) are coplanar.
11. What’s an equation for the unique plane that contains the points \((1,3,2)\) and \((3,-1,6)\) and \((5,2,0).\)
12. At what angle do the planes \(x+y+z=1\) and \(x-2y+3z=1\) intersect? What’s an equation for the line at which those plane intersect?
13. The planes \(10x+2y-2z=5\) and \(5x+y-z=1\) are parallel. Compute the distance between them.
14. Calculate the distance between these skew lines.

    \((x,y,z) = \bigl(1+t, -2+3t, 4-t\bigr)\)

    \((x,y,z) = \bigl(2t, 3+t, -3+4t\bigr)\)

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 ten.
   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 ten.
2. Recall the volume of a tetrahedron is ALSO ⅓ the distance from a point to a face times the area of that face.
3. 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 \(D?\)
4. Given two parallel planes \( Ax+By+Cz = D_1\) and \( Ax+By+Cz = D_2,\) find a formula for the shortest distance between them.
5. 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
6. Devise a formula for the distance from a point \(P\) to a line \({\bm{v}t\!+\!Q.}\)
7. Devise a formula for the distance from a point \(P\) to a plane \({Ax\!+\!By\!+\!Cz\!=\!D.}\)
8. Devise a formula for the distance between the lines \({\bm{u}t\!+\!P}\) and \({\bm{v}t\!+\!Q.}\)
9. 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?
10. 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.

11. 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?
12. 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.