Quandaries and Queries


G'day Helpers,

I have two questions that I just cannot visulise and dont know how to get started on them..!

1. Ship X is sailing North at 4m/s and Ship Y is sailing due East at 3m/s. A sailor on Ship X climbs a vertical pole at 1/2 m/s. What is the velocity of the sailor on Ship X relative to an observer on Ship Y? State it's magnitude and direction.

Not sure where to start with this one...

2. Show, using vectors, that for any tetrahedron, the segments joining the midpoints of the opposite edges are concurrent.
i know what a tetradedron is, and what I am proving but do I solve simultaneously for a specific case?? AH HELP!



I will respond to the second problem.

A good way to visualize this (and it can then be turned into vectors if that is the 'required vocabulary') is to realize the point in question is the center of mass of equal weights at the four corners of the tetrahedron.

If you have four such equal weights, you can:

balance two of them along one (shared) edge - at that midpoint,
balance the other two along the 'opposite' shared edge at that midpoint
balance these two double weights at the midpoint of the segment joining the two midpoints.
This final point is the centroid.

The same centroid is created for each of the opposite pairs of edges, so they have this point in common.

Note- as a bonus, you could balance three of them at a centroid of one face, then balance that with the fourth weight to get another line through the centroid, splitting this new segment in a ratio 3:1. There are actually six distinguished segments going through this centroid!

To observations about this - if you think with transformations (as people in geometry like to do).

With Affine transformations, all tetrahedra (which are not flat in a plane) are the same as the regular tetrahedron - and midpoints go to midpoints. If you check this fact for the one example of the regular tetrahedron, then it must be true for all tetrahedra that are not all flat in a single plane.

With projections, this is also true for the plane image of the tetrahedron. You can then explore this projection with tools like Geometer's Sketchpad or Cabri Geometrie.

Finally, a comment on using balancing weights. Statics is an old tool for doing geometric reasoning. It has been used for centuries and gives some very nice insights. For example, for a plane triangle, this gives the intersection of the medians. Generalized to distinct weights it gives Ceva's Theorem (and apparently is the way that Ceva first understood this theorem).

Walter Whiteley
York University


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