Objectives
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Practice visualizing vector addition
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Use vectors without explicit coordinates
Section 5 Worksheets
Worksheet 5.1 A Geometric Prelude
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 5.1.
In this exercise, we’ll use vectors to show that the medians of any triangle (Figure 5.2) 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 5.3 from the previous page is reproduced for your convenience.
2.
Show that \(\vec{M}_{1} + \vec{M}_{2} + \vec{M}_{3} = 0\text{.}\)
Hint.
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.
