The Dot Product, Cross Product, and Vector Geometry

Exercises

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?
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 your terminal point?
If you travel along a vector \(\langle 2,3,-1 \rangle,\) and end at the terminal point \((-1,7,3)\) what are the coordinates of your initial point?
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:

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

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

\( |\bm{u}| \)

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

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

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

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

\( \bm{u} + 17\mathbf{j} \)
What are the values of the following dot products?

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

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

\(\displaystyle \left(\mathbf{i}+2\mathbf{j}-3\mathbf{k}\right)\cdot\left(2\mathbf{j}-\mathbf{k}\right) \)
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?
What is the angle between the vectors \(\bigl\langle 2,2,1 \bigr\rangle\) and \(\bigl\langle 5,-3,2 \bigr\rangle?\)
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 \(\bm{v}?\)
What are the vector results of the following cross products?

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

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

\(\displaystyle \left(\mathbf{i}+2\mathbf{j}-3\mathbf{k}\right)\times\left(2\mathbf{j}-\mathbf{k}\right) \)
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?
Show that the vectors \(\langle 1,4,-7 \rangle\) and \(\langle 2,-1,4 \rangle\) and \(\langle 0,-9,18 \rangle\) are coplanar.
What’s an equation for the unique plane that contains the points \((1,3,2)\) and \((3,-1,6)\) and \((5,2,0)?\)
What’s an equation for the unique plane that crosses the \(x\)-axis at \(7,\) crosses the \(y\)-axis at \(-3,\) and crosses the \(z\)-axis at \(10?\)
Consider the plane given by equation \(-x+3y+3z = 2.\) Write down the formulas for the lines at which this plane intersects the three coordinate planes, and using those equation sketch an accurate picture of plot of that plane.
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?
The planes \(10x+2y-2z=5\) and \(5x+y-z=1\) are parallel. Compute the distance between them.
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

Suppose that points \(P\) and \(Q\) and \(R\) are the vertices of a triangle. What must be true of the vector \(\overrightarrow{PQ} +\overrightarrow{QR} +\overrightarrow{RP}?\)
Consider the line given parametrically by the equation \(\bigl(3,2,1\bigr) + \bigl\langle -1,4,3\bigr\rangle t.\) Since its direction vector is not parallel to any of the coordinate axes, it must intersect each of the three coordinate planes. For each coordinate plane, compute the acute angle at which the line strikes that plane.
Sketch the vectors \(\bm{u} = \bigl\langle 1,2 \bigr\rangle\) and \(\bm{v} = \bigl\langle 3,1 \bigr\rangle\) and \(\bm{w} = \bigl\langle 7,6 \bigr\rangle\) in \(\mathbf{R}^2\) and note visually that they are linearly dependent i.e. there must be some scalars \(a\) and \(b\) such that \(\bm{w} = a\bm{v} + b\bm{u}.\) Once you’ve convinced yourself of this with a sketch, calculate the precise values that \(a\) and \(b\) must be.
Show that for vectors \(\bm{u}\) and \(\bm{v}\) with angle \(\theta\) between them, that \[ \bigl(\operatorname{proj}_{\bm{u}}(\bm{v})\bigr) \cdot \bigl(\operatorname{proj}_{\bm{v}}(\bm{u})\bigr) = (\bm{u}\cdot\bm{v})\cos^2(\theta)\,. \]
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.
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 \(D?\)
Given two parallel planes \( Ax+By+Cz = D_1\) and \( Ax+By+Cz = D_2,\) find a formula for the shortest distance between them.
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
Devise a formula for the distance from a point \(P\) to a line \({Q\!+\!\bm{v}t.}\)
Devise a formula for the distance from a point \(P\) to a plane \({Ax\!+\!By\!+\!Cz\!=\!D.}\)
Devise a formula for the distance between the lines \({\bm{u}t\!+\!P}\) and \({\bm{v}t\!+\!Q.}\)
For a line containing points \(P\) and \(Q\) and another point \(X\) not on that line, show that the shortest distance to that line from \(X\) is \[ \frac{\left|\overrightarrow{PQ} \times \overrightarrow{PX}\right|}{\left|\overrightarrow{PQ}\right|}\,. \]
For a plane containing points \(P\) and \(Q\) and \(R\) and another point \(X\) not on that plane, show that the shortest distance to that plane from \(X\) is \[ \frac{\left|\overrightarrow{PQ} \cdot \bigl(\overrightarrow{PR} \times \overrightarrow{PX}\bigr)\right|}{\left|\overrightarrow{PQ} \times \overrightarrow{PR}\right|}\,. \]
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?
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.
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 (N 5° E).
1. What is the wind velocity?
2. In what direction should the pilot have headed to reach the intended destination?