SA Worksheets

Worksheet 34.1 A Geometric Prelude
A4 US

Objectives

Practice visualizing vector addition
Use vectors without explicit coordinates

This two-page worksheet was generously donated to the sample article by Dave Rosoff at a CuratedCourses workshop in August 2018. It has the default (skinny) margins.

It was known to Euclid, and probably earlier, that the midpoints of the sides of any quadrilateral all lie in the same plane (even if the vertices of the quadrilateral do not). In fact, these midpoints are the vertices of a parallelogram, as pictured in Figure 34.1.

Figure 34.1. The midpoints of the sides of a quadrilateral are the vertices of a parallelogram.

Figure 34.2. The sides of a triangle presented as vectors.

Figure 34.3. The medians of the triangle are \(\vec{M}_1\text{,}\) \(\vec{M}_2\text{,}\) and \(\vec{M}_3\text{.}\)

In this exercise, we’ll use vectors to show that the medians of any triangle (Figure 34.2) intersect at a point. Recall that medians are the lines connecting the vertices of the triangle to the midpoints of their opposite edges, as in the figure. We’ll do this in a few steps.

1.

What is the value of \(\vec{A} + \vec{B} + \vec{C}\text{?}\)

Figure 34.3 from the previous page is reproduced for your convenience.

Figure 34.4. The medians of the triangle are \(\vec{M}_1\text{,}\) \(\vec{M}_2\text{,}\) and \(\vec{M}_3\text{.}\)

2.

Show that \(\vec{M}_{1} + \vec{M}_{2} + \vec{M}_{3} = 0\text{.}\)

Hint.

Use Worksheet Exercise 34.1.1.

3.

To show that the point \(P\) exists (as the common intersection of the \(\vec{M}_{i}\)), show that

\begin{equation*} \vec{A} + \frac{2}{3} \vec{M}_{3} = \frac{2}{3} \vec{M}_{2} = \fillinmath{\frac{2}{3} \vec{M}_{1}-\vec{C}}\text{.} \end{equation*}

4.

If you have time, try to devise a vector proof of Euclid’s result presented at the beginning of the workshop. Recall that a parallelogram is a four-sided polygon whose opposite sides are parallel.

Wrap-up.

It’s possible to do interesting things with vector arithmetic in a coordinate-free way: we didn’t specify an origin, or any entries of any vectors in the examples.

Worksheet 34.2 Networks Worksheet
A4 US

Basic laws for electrical circuits.

This two-page worksheet was generously donated to the sample article by Virgil Pierce at a CuratedCourses workshop in August 2018. It has default (skinny) left and right margins, but we have specified longer top and bottom margins, with the top being the larger of the two.

Theorem 34.5. Ohms Law.

The current through a resistor is proportional to the ratio of the Voltage to the Resistance

\begin{equation*} I = \frac{V}{R} \end{equation*}

Or for our purposes

\begin{equation*} I R = V \end{equation*}

Theorem 34.6. Kirchoffs Current Law.

The sum of the currents in a network meeting at a point is zero.

\begin{equation*} \sum_{k=1}^n I_k = 0 \end{equation*}

Example 34.7. Kirchoff’s Current Law.

For the circuit below \(I_1 + I_2 = I_3 \text{.}\)

Theorem 34.8. Kirchoffs Voltage Law.

The sum of the voltages around any closed circuit (or subcircuit) is zero.

\begin{equation*} \sum_{k=1}^n V_k = 0 \end{equation*}

Kirchoffs Current Law and Kirkoffs Voltage Law combined with Ohms Law gives for any circuit of resistors and sources a linear system that may (or may not) determine the currents.

1.

For the simple network pictured, calculuate the amperage in each part of the network by setting up a system of linear equations for the amperages.

2.

Compare it with a parallel circuit network. Calculate the amperage in each part of the network by setting up a system of linear equations for the amperages.

3.

Now for a more complicated network. Calculate the amperage in each part of the network by setting up a system of linear equations for the amperages.

4.

Now generalize these ideas to a context outside of electrical circuits. Consider the network of streets given in the diagram (with one-way directions as indicated).

A traffic engineer counts the hourly flow of cars into and out of this network at the entrances. They get (EB = East Bound; WB = West Bound):

Table 34.9. Estimated hourly traffic flow for the road network

	EB Winooski	WB Winooski	Shelburne St	Willow	Jay
into	50	400	0	10	50
out of	55	390	20	15	30

Use a variable for each segment inside of the network and set up a system of linear equations restricting the flow. Solve the system. Note that you should not get a unique solution as traffic should be able to flow through the network in various ways.

Worksheet 34.3 Worksheet
A4 US

This is a mock one-page worksheet for testing purposes. We have specified an overall margin just slightly less than the default.