Give me a Vector Victor
If you've ever watched the comedy film "Airplane" there's a memorable cockpit scene where the Captain is called Roger and the Navigator Victor. It goes something like this:
Captain: "Give me a Vector Victor"
Navigator: "Roger, Roger" and so it goes on...
So, what is a Vector? It's a thing which has both size and direction. So what's that mean? Well, think of the speed of a car being 70 miles per hour. Speed is not a vector as it only has size (10mph is a small size and 70mph a much bigger size), things with only size are called Scalars.
Now, think of a car doing 70mph in an North Easterly direction (i.e. a bearing of 45o from North). We are now thinking of a speed and direction, this is called velocity and is an example of a vector.
We can show what this velocity looks like with an arrow:
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This sort of diagram can be used as a model to show an image of a vector. The length of the vector arrow is used to represent its size (in our case 70mph). The direction of the arrow is measured as we would any other direction (by the angle, which is 45o in our case). |
Just to make things doubly painful
when dealing with vectors, there are various ways of indicating that something
is a vector. Here are 3 ways of representing the same thing in books:
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The last way shown to indicate a vector
(by using bold type) is obviously tricky to do if you are writing by hand. Here
the convention is normally to use a squiggly line underneath the letter to indicate
that you are talking about a vector e.g.
Now you know how vectors are written, we can see how vectors are added and subtracted.
If you are at Heathrow and want to fly to Edinburgh, you can either go straight there or you can make two flights by stopping off half-way at Manchester. Each flight can be shown as vectors:
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Here the flight from Heathrow to Manchester is shown as vector a and the flight from Manchester to Edinburgh is shown as vector b. Whether we go via Manchester or have a direct flight we still end up at the same destination. Hence we can say that the vector for the direct flight from London to Edinbugh = the flight from Heathrow to Manchester + the flight from Manchester to Edinburgh. |
When subtracting vectors it helps
to think of doing e.g. a + (-b) rather than a - b
as can be seen in the diagram below:
You may read about the parallelogram method - forget this as it's the same stuff for adding and subtracting vectors as above only more complicated.
We found earlier that the length of a vector indicates its size and is called the magnitude. How do we find out what the magnitude is?
Remember that great old hippy Pythagoras? That's all there is to it! All you need to do is turn the vector into the hypotenuse of a right-angled triangle.
There are a couple more things you need to know about vectors:
