This is a section full of worksheets. Each is a division of its own, via the <worksheet> element. This is an optional <introduction> to the current <section>. In practice you might want to rip out all the worksheets of an entire book and bundle them up as an βactivity book.β
If you make PDF output you will notice an increased amount of control over layout. Also, if the publication file elects LaTeX draft mode, then there will be visual indicators of prescribed whitespace.
This two-page worksheet was generously donated to the sample article by Dave Rosoff at a CuratedCourses workshop in AugustΒ 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 FigureΒ 35.1.
In this exercise, weβll use vectors to show that the medians of any triangle (FigureΒ 35.2) 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. Weβll do this in a few steps.
If you have time, try to devise a vector proof of Euclidβs result presented at the beginning of the workshop. Recall that a parallelogram is a four-sided polygon whose opposite sides are parallel.
Itβs possible to do interesting things with vector arithmetic in a coordinate-free way: we didnβt specify an origin, or any entries of any vectors in the examples.
This two-page worksheet was generously donated to the sample article by Virgil Pierce at a CuratedCourses workshop in AugustΒ 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.
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.
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.
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).
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.
Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra.
Integer sagittis dictum turpis vel aliquet. Fusce ut suscipit dolor, nec tristique nisl. Aenean luctus, leo et ornare fermentum, nibh dui vulputate leo, nec tincidunt augue ipsum sed odio. Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.
Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra.
Integer sagittis dictum turpis vel aliquet. Fusce ut suscipit dolor, nec tristique nisl. Aenean luctus, leo et ornare fermentum, nibh dui vulputate leo, nec tincidunt augue ipsum sed odio. Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.
Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra.
Integer sagittis dictum turpis vel aliquet. Fusce ut suscipit dolor, nec tristique nisl. Aenean luctus, leo et ornare fermentum, nibh dui vulputate leo, nec tincidunt augue ipsum sed odio. Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.
Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra.
Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra.
Integer sagittis dictum turpis vel aliquet. Fusce ut suscipit dolor, nec tristique nisl. Aenean luctus, leo et ornare fermentum, nibh dui vulputate leo, nec tincidunt augue ipsum sed odio. Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.
Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra.
Let \({\vec v}_1 = (-4,1)\text{,}\)\({\vec v}_2 = (2,2)\text{,}\)\({\vec v}_3 = (1,2,3)\text{,}\)\({\vec v}_4 = (-2,1,0)\text{.}\) Find the values of the following expressions:
The vectors \(\vec a = (3,9)\) and \(\vec u = (4,2)\) are pictured below. Derive the formula for projection on a line and use it to find the projection of \(\vec a\) on the line spanned by \(\vec u\text{.}\) Also compute the length of the residual vector.