Are vectors just a mathematical idea, or are they of any practical use? Every time you fly anywhere, it’s because of vectors that you end up in the right place! Here’s why.
What is a vector?
A vector represents any quantity that has both magnitude and direction. So, vectors can be used to represent displacement (ie change of position), velocity, force – but nor distance, speed or energy. For example, if the distance between two towns along a winding road is 120km, and it takes me 2 hours to drive, my average speed is 60 km/hour. However, if the second town is 100km due South of the first town, then my displacement is exactly that, and my average velocity is 50 km/hour due South – it is the average amount, per hour, by which my position has changed.
What is vector addition?
Suppose in the previous example I travelled from the second town to a third town 50km due East – what would my overall displacement (change of position) be? The diagram below shows the situation, with the red vector showing my change in position from the first town to the third:
If we represent the 100km South by the vector AB, and the 50km East by vector BC, then the overall displacement, 111.8km on a bearing of 153° (found using Pythagoras and Trigonometry), vector AC, can also be written as
AB + BC. Think of the sum in terms of a “path”: if AB is the direct path from A to B, and BC the direct path from B to C, then AB + BC = AC, the direct path from A to C. Note that you’re not adding the actual distances together, but….
Why use the word “addition”?
Because if you use the x – y component form for vectors, then addition is involved. For example, suppose I add the vectors:
The vector diagram for this addition looks like this:
and we can see that the x and y components of the resultant vector are simply the sum of the components of the original vectors.
Practical application of vector addition
If you’ve flown, you may have heard the pilot announce that because of a strong tailwind, the plane will arrive ahead of schedule For example, a plane travelling through the air at 500 km/hour with a tailwind of 50 km/hour will actually be travelling over the ground at 550 km/hour; similarly, if there is a headwind of 50 km/hour, the speed over the ground would be 450 km/hour. We can illustrate these situations, for a plane heading due East, using vector addition:
The beauty of using vectors is that we can use the same diagram whatever direction the wind is blowing. For example, if the wind is from the the southwest:
From this diagram we can see that the plane is travelling faster than 500 km/hour, but is no longer heading in the right direction. We need to draw a new diagram to work out the correct heading:
The plane must fly at 094° (using the sine rule). 4° difference may not sound much, but on a 2000km flight the error in position would be 140km – the passengers would not be too happy!
The same principle applies if you are rowing a boat across a river. If you simply aim for the point you want to get to, you will end up downstream from there. You need to point the boat slightly upstream to take account of the flow.