Learn Vectors in 5 Minutes
You're sitting in a room the shape of a cube. In the room is a square table and on the table is a square sheet of paper. I touch the sheet of paper with the tip of my pen to produce a dot. Then I ask you a question..
What's the location of the dot?
You go to work. First you pick any corner of the sheet of paper and mark it as the origin. You rotate the paper so your origin is at the bottom left. Then you simply measure the distance to the dot along the horizontal and vertical edges and you get two numbers, let's call them x and y. Then you tell me the location of the dot is (x,y).
There's two important things to note about this simple measurement. First, the result requires two numbers. You try hard, but no matter how hard you try you cannot specify the location of the dot with just one number. Second, your result is only valid relative to the sheet of paper. For example, if I measure the location of the dot relative to the sides of the table I get a totally different result. And if I measure the location of the dot relative to the sides of the room I not only get a totally different result but I need three numbers (x,y,z) to specify the location!
Now, on the sheet of paper, draw a straight line from the dot to your corner origin. That's a vector! Notice that it has a length and a direction. The length is often called the magnitude, which is why most books define a vector as "an entity that has both magnitude and direction". I think that definition is a bit dry, and I prefer the paper/dot experiment.
Now imagine I measure the location of the dot relative to the table. If the paper is moved on the table my measurement will change, but yours will not. So if the paper moves around on the table I have to do a lot of new measurements, but you don't have to do any!
What about the length of the vector? The length can be specified by just one number and for this we both get the same answer. That's because the length is independent of the reference frame. Things like this are called invariants.
Vectors are incredibly useful in physics, where they describe things that have magnitude and direction such as force and velocity. In fact, one of the most fundamental statements in physics is this..
"If the velocity of an object changes a force must be acting on the object."
Of course, since velocity is a vector it can change in several ways: its direction may change, its magnitude may change, or both may change, and a force will appear in any of these situations.
Since vectors are mathematical objects can we define operations on these objects? Of course. One of the beauties of mathematics is that you can define anything you want! So what about addition, can we add two vectors? Yes. If A=(a1,a2,a3) and B=(b1,b2,b3) are two vectors then we might define addition as..
and this definition turns out to be very useful.
What about multiplication? This is interesting because there are two definitions of multiplications that are useful. One is called the dot product and one is called the cross product.
The dot product of two vectors is just a number..
Most of the usefulness of the dot product comes from the simple fact that for any vector A.A=(length of A)^2
The cross product of two vectors is a vector, and when you first see the definition it looks strange and complicated..
It takes the components of A and B and totally mixes them up! But it's not really that strange. It turns out that the cross product is a vector perpendicular to the plane containing A and B. So for two vectors on our sheet of paper the cross product would be a vector pointing directly up out of the paper. Its length is (length of A)*(length of B)*sin(q), where q is the angle between A and B. Despite its apparent complexity, the cross product turns out to be incredibly useful in all sorts of applications.
Now let's do something that mathematicians love to do - generalize. Can we generalize the concept of a vector? Yes. There are probably many ways to do this, but one is obvious. When we measured the location of the dot relative to the sheet of paper we needed two numbers (x,y). That's because the paper surface is 2-dimensional. When we measured it relative to the room we needed three numbers (x,y,z). That's because the room is 3-dimensional. So what about this..
It has n components. It's a n-dimensional vector!
Content written and posted by Ken Abbott firstname.lastname@example.org