Tuesday, March 3, 2009

Vectors: Second Thoughts

In developing a vectorial algebra, we introduce the idea of unit vectors i, j and k. This initially can seem quite odd and counter intuitive - why do we need to impose "directions" onto three dimensional space ? What is wrong with x, y and z (Cartesian co-ordinates) ?

I think this type of ques ion is best answered by "doing", that is, the whole point of using unit vectors to explain relationship in space become obvious when you start to using these quantities but I will do my best to justify this choice (and like a lot of mathematics, these symbols represent an intellectual choice i.e. we choose this particular abstraction to help us develop ideas) from the beginning and independent of this experience ("a priori" is the Latin for this concept).

Lets have a go .... imagine you are interested in analysing the wind patterns over Melbourne. Your raw data is wind speed and direction data collected from weather stations dotted around Melbourne. Imagine, this data is collected continuously but "average" data is collated every five minutes. How would you represent and analyse this data ? Would you express the changes in wind direction between the various weather stations through references to their various map c0-ordinates (latitudes and longitudes, or even the Melway's grid reference system) or would you express the vectors at each location ? I think the asnwer to that question is obvious but I will let you think it through ! How would you resolve the wind speeds and directions in the areas between weather stations ? Let me again suggest that using vectors to resolve this issue will be alot easier than trying to use map based physics.

It was this type of analysis that influence physicists, mathematicians and engineers to shift to a vectorial description of the world some hundred and fifty years ago. It was simply to cumbersome to try to using a Cartesian type "map" system to analyse complex physical problems.

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