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A lesson in vectors

There are some concepts that are so useful that we should really try to push them out to everyone.  One of these is the notion of a 'vector'.  Humans have a very intuitive grasp already of what vectors are, and how they can be used, although they often don't realize it.

I find the following example very useful when trying to teach this concept deeply and quickly.

Say you're in Manhattan, and you have a GPS device that gives latitude and longitude coordinates.  Manhattan has a perpendicular grid of streets, but they are not aligned with the cardinal directions.  Rather, Manhattan is tilted about 30 degrees clockwise.  As a result, people speak of "uptown" versus "downtown", and "cross-town" in approximately "East" and "West" directions.

If I ask directions from a person on the street, they might tell me to walk four blocks uptown, then three blocks cross-town West, to arrive at my destination.  The fact that I'm walking on a roughly 2D surface means that I need two numbers to get where I'm going.  In an empty desert, this could involve pointing in a direction and saying, "Walk 1000 paces".  This two-dimensional instruction involves an angle and a radius.  On a grid, we need to think in terms of Cartesian numbers, which is why I am advised to walk two perpendicular distances.  Both of these instructions are vectors, but they use different actual numbers to tell the same story.

Now, if I want to use my GPS to figure it out for myself, I'm going to need to do some mental rotation.  A destination that is straight North will require me to go "uptown" some distance, then "cross-town West" some distance.  If I figure this out, I have just done some matrix algebra...whether I recognize it or not.

The author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the author.

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