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Spotting the Loophole

Age 14 to 16 Challenge Level:

Why do this problem?

This problem encourages students to use visualisation to help them to spot a solution which can then be verified using algebra. It shows students the power of using visual representations to solve vector problems, often the quickest route to a solution.


Possible approach

This could form a short introduction to work on vectors.
Display the example grids on the board, showing the three vectors forming a closed loop and the four vectors on the right which do not have a zero sum.

Then challenge students to find any closed loops in each of the three grids.
When they think they have spotted a closed loop by eye, they should verify algebraically that it is indeed a closed loop.


Key questions

What can you say about the horizontal components of the vectors in a closed loop?

And what about the vertical components?


Possible support

Students may find it helpful to draw the vectors on squared paper.


Possible extension

Vector Walk begins to explore properties of vectors and combining two basic vectors to reach points on the coordinate grid.