Getting Back Home Animation
From guessing and/or analysing the system of linear equations, we found that one way to travel in a loop was to ride v_1 forward for two hours, v_3 forward for an hour and v_2 in reverse for an hour. We also noticed that if we had found one solution, that scalar multiples are also solutions.
In fact, this means that the only important part is the ratio of the different times.
Plotted here is a picture of riding v_1 for length of time 2a, riding v_3 for length of time a, and riding v_2 in reverse for length of time a. You change the value of a to see different solutions plotted. (You can also swivel the picture around to get a better view.)
Here are some of the great questions I heard today. Please choose at least 3 of them to think about over the weekend:
1) Is there more than one way to get back home?
2) Can you get back home using only a pair of these vectors?
3) Does order of traversing the path matter? (for example, can we do v_2 in reverse then v_1 and then v_3?) 4) Can you write v_2 as a linear combination of v_1 and v_3?
5) Is v_3 in the span of v_1 and v_2?
Several groups found a solution of c_1 = 2c_3 and c_2=-c_3 to the system of linear equations. How does this help us find different ways to get home?