Math CentralQuandaries & Queries


Question from Jasmine, a student:

Bob is traveling due north he then turns left 45 degrees followed by four 20 degree turns to the right

What is his new heading if he needed to turn so his heading was SSW what direction and what angle should he turn?

One interesting aspect of this question is:
Is Bob on the earth (a sphere) or on a plane?

If he is on the earth (sphere), then the location on the earth makes a big difference, as does the length of the paths taken (relative to the radius of the earth/sphere).

So I imagine that the answer requires you to pretend he is walking on an infinite plane!

Now, the problem has recognized that the distance travelled no longer matters.
So - you might as well assume he does not travel at all. Just turns on the spot!
(This is the kind of simplification that can happen when you play with the problem in your mind - or in something like Geometer's Sketchpad. Thinking is often a lot easier than doing a lot of extra calculations.)

Draw a unit circle centered at Bob. Put down North (probably at the very top). Mark off the three 20 degree turns to the right (clockwise). Find that point on the circle.

Now mark the direction SSW onto the same circle. My guess is you can mark each of these angles onto the circle, and do a subtraction(s) to find out the difference in directions. (Note that, at the very end, he could turn clockwise or counter clockwise to get there, so there are TWO answers!

Walter Whiteley

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