Vectors made easy !

The unit vector notation used in Engineering mathematics is wonderfully simple and powerful.

Imagine we have a position in the x/y plane, lets call it P, and we want to form a vector from the origin to this point. We call that position vector OP.

Lets say that P is located at x=2 and y=3, now we can draw a line from the origin to P and put an arrow head along it. We have a vector.

Now lets define the unit vector (i.e. one unit in a particular direction) in the x direction as i and the unit vector in the y direction as j.

We can now say OP = 2i + 3j.

Lets try a few things ....

What is the magnitude of this vector ?

If we draw a right angled triangle from OP, we can quickly see that the magnitude of OP must be equal to the square root of (2 squared + 3 squared) = sqrt(13).

What is the unit vector of OP ?

It must be OP/sqrt(13) i.e. one unit in the direction of OP.

What about if we want to add another vector (OQ = 4i + 1j) to OP ?

OP + OQ = (4 + 2)i + (3 + 1)j = 6i + 4j.

What if I want to find the vector QO ?

QO = -OQ = -(4i + 1j) = -4i-1j

What about if we want to find the vector PQ ?

We use the head to tail rule and say PO + OQ = PQ = -(2i + 3j) + (4i + 1j) = 2i - 2j.

This approach makes the whole problem of manipulating vectors easy and more powerful than constantly referring to angles and scalar qauntities.

Torque about Cross Products

Archimedes is accredited with saying "Give me a place to stand and I will move the earth". This statement reflects his profound understanding of the lever principle which lead to the invention of block and tackle pulley system, which apart from saving many backs since it's inception, represents a significant milestone in the development of machines. Levers and pulleys illustrate the mechanical advantage gained by applying a force over a distance. In vectorial form, we write the relationship as:

M = F x r = "moment" vector or "torque" provided by applying a force over a distance.

where F is the force vector and r is the postions vector, and the "x" symbol represents the operations of "cross products" or "vector products". "Torque" and "moments" are imporant mechanical concepts to understand, as they underpin our understanding of machines. The starting point is the simple lever. Appreciate how a lever works (whether it be a spanner or a see saw) and you can start to appreciate the notion of torque. A "moment" is in effect the same as "torque", except the later is generally reserved for scenario's where the moment is induced by circular motion (e.g. the torque induced in an axle by a force being applied to a wheel).

Further analysis also shows:

M = F x r = Fr sin k n

where k is the angle between F and r, F and r are the magnitudes of the two vectors, and n is the unit vector perpendicular to the plane containing F and r.

Very conveniently F x r = det( i j k, F1 F2 F3, r1 r2 r3)

(if you haven't done matrix algebra see http://en.wikipedia.org/wiki/Determinant for a quick explanation)

which is the same, as saying:

F x r = (F2r3-r2F3)i - (F1r3-r1F3)j + (F1r2-r1F2)k

where F = F1i + F2j + F3k and r = r1i + r2j + r3k

Thus, the cross product functions allows important mechanical calculations to be performed and for vectors perpendicular to a particular plane to be easily calculated.

Dot Products ?

Dot products or "scaler products" are the first significant manipulation we learn to use for vectors after the famous "head to tail" rule. Unlike the "head to tail" rule of vector addition and subtraction, dot products appear initially to be somewhat obscure, however, they have both a signficant geometrical and physical meaning.

Geometrically, the definition of a dot product allows the angle between to two vectors to determined quite rapidly using:

a.b = ab cos x = xaxb + yayb + zazb

a = xai + yaj + zak and b = xbi + ybj + zbk

Re-arranging

cos x = (a.b)/(ab)

where the bold italic symbols refer to vectors, x is the angle between the two vectors and non-bold symbols refer to the magnitudes of the vector. This is a very straight forward calculation for two vectors that are defined and is much simpler than the comparable cartesian algebraic approach.

The procedure also has a physical meaning, for example, the work done (W) by a Force (F) displacing an object along a vector (r) can be calculated using:

W = F.r

In this case the dot product has a precise a physical meaning .... sounds like a good idea, simple and useful !!

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.

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 http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html .