Sunday, March 1, 2009

Vectors: First Thoughts

The first topic in the Swinburne Engineering Mathematics subject for first year is Vectors. For some of the students, this will be a new topic depending on your background in high school mathematics and physics.

First things first .... what is a vector ?

A vector is quantity that has both size and direction. What does that really mean ? If I ask you how fast your car is going (assuming that you are silly enough to give your old Prof. a lift) and you answer a 100 km/hr, mathematically you have given me an answer that is described as a Speed which is a Scalar quantity (a quantity that has size but no direction). If you answer 100km/hr towards the city along the Monash freeway, you have given both direction and quantity and thus a Velocity (which is a vector).

This Monash freeway brings up some other issues about vectors because as we drive along the freeway, your instantaneous speed is likely to change as you accelerate and de-accelerate. Likewise your instantaneous direction will also change as you drive along the freeway, as the Monash doesn't follow the same compass bearing into the city, for example, it turns quite northerly near Burwood Highway but starts turning westward again as its approaches the Burnley tunnel.

In this very simple example, we can see that the velocity of the car is a function of time and position and the intuitively wise among you can see a very important topic rearing it's head, that is, CALCULUS OF VECTOR FUNCTIONS i.e the mathematics of change for functions that have both magnitude and direction. Because we are nice guys at Swinburne, we don't throw you into that topic straight away, we firstly make sure that you are very comfortable with the mathematics of vectors and the details of calculus before combining the two together. Something to look forward to !

Are vectors important to engineers

Absolutely, vectorial quantities are critical to engineering, they help us understand the complex stress-strain relationships in bridges, the movement of fluids in pipes and channels, the flow of air around an aircraft's wing, the interplay of electrical/magnetic fields in circuits, and numerous other examples. It is difficult to imagine engineering without vectorial analysis .... without this wonderful mathematical tool, we would be left with trying to analyse complex situations with simple addition/subtraction equations and cumbersome manipulations on X-Y co-ordinates. We would be trapped in an endless Year 10 world !! Sounds like hell to me !

How do we get good at vectors ?

The first step, in my opinion, is to be comfortable with simple physical examples before moving into the details of the algebra. This is why the problems 1 to 3 on page 12 of the student notes are important. You need to be a master of these type of problems ("A boat heads off in 20 km/hr in a NE direction with a wind blowing 50 km/hr due south ....) before moving onto the questions that are more algebraic in nature. When addressing these questions, I suggest:

a) working through the examples on page 1 to 5,

b) sketching the problem and trying visualise what the answer would look like,

c) applying the head to tail rule, being careful to distinguish between problems where (i) the resultant vector is not known (therefore, you add the two vectors head to tail) and (ii) where the resultant vector is known (there, you will need to subtract vectors to work out the vector that is missing), and

d) looking at your answer and asking yourself "Does that make sense ?".

As always, a sense of humour, determination and willingness to be challenged will help.

Historical Note:
Vectors began emerging as a distinct mathematical idea during the 19th century through the ideas of Wessell (1745-1818), Argand (1868-1822), Gauss (1777-1845) and but the first really thorough treatment of the concept is generally acredited to William Hamilton (1805-1865) who developed a form of vector algebra based on manipulating "quaternions". The function "H" which is used to express the change with time of the condition of a dynamic physical system (e.g. a set of ball flying in the air), is named in his honour. Interestingly, this development of vector algebra is tied up with the development of another important topic mathematics, that is, complex numbers. These developments are well described in the book "Unknown Quantity: A Real and Imaginary History of Algebra" by John Derbyshire (Alantic Books, London, 2006) or if you want the quick story, go to .

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