# Vector complexity?

1. Feb 2, 2014

### A.J.710

I am currently learning vectors in physics. Things such as adding, subtracting, multiplying and converting between polar and Cartesian. I was just wondering how complex this concept really is. I am completely confused when I look at it but when the teacher tries to explain it I feel like it's an easy concept and I just need to sit down and play around with the numbers until it clicks.

Is this really a hard concept to grasp because it is really confusing me...

2. Feb 3, 2014

### Simon Bridge

It can be tricky when you look at the algebra - but it is really just lots of different ways of representing an arrow, and of representing things as arrows.

The important things about arrows is that they have a length and a direction.
That's it for the concept.

3. Feb 3, 2014

### jbrussell93

Are you confused about the concept of a vector? Or is it more about the operations with vectors (cross product, dot product, etc) and what they REALLY mean.

Vectors confused me at first too because for the first time you are forced to consider a DIRECTION in mathematics. It will become clearer the more you practice with and think about vectors.

4. Feb 3, 2014

### A.J.710

I completely understand the concept as in what vectors are. As my professor says "a book-keeping system" for lines. It's just the operations that confuse me a lot.

5. Feb 3, 2014

### jbrussell93

Well, you are half way there then
Addition and subtraction of vectors is pretty straight forward. If you don't agree, then just play around with adding and subtracting them geometrically/graphically to build up the intuition.

For multiplication, things get a bit trickier. There are 2 ways to multiply two vectors: Dot product produces a scalar and Cross product produces another vector. Other than the act of actually computing them, the geometrical intuition is very important.

Dot product: I like to think of the dot product as a measure of how 'parallel' two vectors are. Mathematically, you can understand this by the formula A.B=ABcos(theta) where theta is the angle between them. As the angle gets smaller, theta gets closer to zero and cos(theta) approaches its maximum. Therefore, two vectors are exactly parallel when their dot products are maximum. An important thing to note is that the dot product of two vectors is a scalar

For a more physical interpretation, think of it as the projection of one vector onto another. For example, think about a tilted lamp post sticking out of the sidewalk. The lamp post is a vector "A" pointing towards the sky and a vector "B" is pointing along the sidewalk away from the post. Lets say it is noon and the sun is shining directly overhead, casting the lamp's shadow on the sidewalk. The dot product would be sort of analogous to the length of this shadow.

Cross product: In contrast to the dot product, I think of the magnitude of the cross product as a measure of how 'perpendicular' two vectors are. The argument is the same as the one above for the dot product but with sin(theta) instead of cos(theta). When two vectors are exactly perpendicular (90 degrees), the magnitude of the cross product is maximum. But this is only the magnitude of the vector. The important thing to note is that the cross product produces a vector who's direction perpendicular to BOTH vectors being multiplied.

Another interesting fact about the cross product is that the magnitude of the cross product is equal to the area of the parallelogram determined by the two vectors.

These are just a couple of things to keep in mind though I'm sure your teacher has made these points. Good luck

6. Feb 5, 2014

### Simon Bridge

Hmmm... well it's tough to go from the first bit to the second, yes.
While vectors can be used as a book-keeping system for lines, I'm not sure that's a terribly helpful way to think about them. i.e. how would the dot and cross product of two lines work? (They can do...)
Probably better to go back to them being a way of describing arrows.

Without the direction part you just have a line segment ... which is what a normal number is at heart: a quantity without a direction. The standard operations on normal numbers are multiplication and addition.

When you have to take account of a direction, these operations become less straight forward.
It is usually more helpful to go from the meaning to the operation rather than the other way ...

i.e. you walk a distance a in one direction and another distance b in another direction and you want to know how far away c you'll be from the start... then that is vector addition, and you write that as $\vec c = \vec a + \vec b$.

That takes care of addition - what of multiplication?
The dot and cross product are two different ways that the concept of multiplication could apply to something with vectors.

i.e. you want to find the area of something you multiply two lengths together. The approach works if the lengths are in different directions: so here you have the idea of a length and a direction so vectors should be useful here. This is where the cross product comes from ... however, there are other effects of taking the cross-product. i.e. it also includes the concept of being perpendicular. The cross product of two vectors is another vector that is perpendicular to both of them.

The operations are useful because they provide a shorthand for a bunch of related ideas.

The way math is usually taught, you start with the definitions and then get to do a lot of exercises putting the definitions to use. As a result, the whole thing looks unmotivated. Why not have one operation for "find the area" and a different one for "find the perpendicular vector" etc.? The reason is that it is actually simpler this way. The trick is to relate the uses to the operations.

At this point I cannot help clear your confusion about them further without some examples of where you get confused.

Last edited: Feb 5, 2014