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# Vector examples

Visually understanding basic vector operations. Created by Sal Khan.

## Want to join the conversation?

• In the above example, R1 is one dimensional, R2 is two dimensional, and both are represented by x and y axis; likewise, R3 is three dimensional; thus, they are forming x, y and z axis, respectively. But how to represent beyond that? E.g., R4, R5, etc., because if you're saying that we will consider axis till R infinite, then it kind of forms an imaginary sphere of axis surrounding the origin at (0,0,.. ); but even that representation is 3 dimensional, (as of those new R's are contained within the x, y, z axis, and are not increasing any dimensions). So how to plot beyond the three dimensions?
• Another way of imagining a 4D plot is to think of a three-dimensional space, where EVERY point in the space is filled with a color that represents the value of the 4th component of each 4-tuple. Or imagine temperatures instead of colors if that works better for you.

But generally speaking, yeah, it's basically impossible to directly represent spaces higher than R^3 in any kind of graph that is directly perceivable to our senses. That's one of the limitations of living in a three-spatial-dimension universe. You can sort of imagine them, using various tricks like imagining three-dimensional "slices" being "stacked up", imagining colors or temperatures to represent the fourth quantity, et cetera, but you'll never really imagine what 4D is like.
• why are vectors written in column form?
• because when you have to add or substract one vector from another its easier if you see both numbers you have to add or subtract next to each other. e.g. if you had a vector like this: v(2;4) and you have to add to v(-4;5). You have to look from one side to another watching clearly wath number you have to use. Instead if they are in columns you have the number right next to the other. Much easier.
• At , shouldn't it be (x1-1, x2+2)?
• You are right. I did not notice that at first.
• He says that R^2 is a bigger space then R^1, how is this possible when they're both infinitely large?
• Good question,

There are many ways to compare how large two things are. In this case we might say that R^2 is larger than R^1 because R^1 fits inside R^2 but R^2 does not fit inside R^1.

This is comparable to saying plane is larger than a line you might draw on that plane. Both have an infinite number of points but one contains the other.

We compare numbers by their "value", we can compare polygons by their "area", and we can compare vector spaces by their "dimension". We can also compare these objects in other ways but these are the ones that are the most useful.
• since when you add and subtract vectors, it sometimes forms a triangle, can't we use some basic tric to manipulate other values of the "triangle" and/or redefine trig values in terms of vectors?
Also, I know that we usually assume vectors originate from the origin, eg. (0,0), but is it ever important to denote a vector that originates from another point in space and how would one denote that vector?
• (I only know the answer to the first part of your question). Yes you can show a vector using trigonometry, you say < ||v|| , ∠θ > where the first one is how long the vector is, the second one is how far the vector is rotated counter-clockwise using the positive x-axis as 0. I don't know how (if there is a way) how to use that notation in 3 or higher dimensions.

As an example, let's say I have the vector < 2 , 3 >. That would be a triangle of base 2 and height 3 so the magnitude is the hypotenuse, √(13). Arctan(3/2) = ~56.31°, so the vector would be <√(13) , ∠56.31°>

http://www.wolframalpha.com/input/?i=vector+%282%2C3%29
• How are tuples, vectors and sets different from each other? Or what is the definition of each one and how do vary from each other? Thanks!
• An n-tuple is almost like slang for a list of length n, which looks like (x1, x2, ... , xn). So a 2-tuple could look like (x1, x2) which is a list of length 2, or an ordered pair.
If you can write a set of vectors in such a list then you may call it a tuple.

Here is a reasonable explaination if you need it :)
(Beginning of chapter 1 if it doesn't automatically scroll to it)
(1 vote)
• Okay, R^n means that there is a n-tuple of real numbers. What about n-tumples with different kinds of numbers? For example: (x1, x2, x3, x4) where x1 is a real number, x2 is an integer, x3 is a natural number and x4 is a rational number. How to write that?
• The root of your question, "What about n-tuples with different kinds of numbers?" is more interesting than your example. It is in fact possible to have n-tuples with different "kinds" of numbers. Each element of the n-tuple vector may be assigned it's own unit. For example, you could define the first axis in terms of inches and the second in terms of miles.

The units of the elements don't even need to relate to each other. You might have the first axis defined in terms of temperature and the second in terms of length. It all depends on what you are trying to represent with your vector.
• what is the difference between linear dependence and in dependence?
• Maybe you were confused thinking he was saying two words "in dependence", although it's actually one word? This is discussed at length in the "Linear dependence and independence" video series here. But, the short answer is that dependence is the opposite of independence.
• At , when applying to a vector in 4 dimensions, why does he use 4a - 2b? I thought he was simply finding the difference between the two ends of the vectors.