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Vector intro for linear algebra

A vector has both magnitude and direction. We use vectors to, for example, describe the velocity of moving objects. In this video, you'll learn how to write and draw vectors. Created by Sal Khan.

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  • starky ultimate style avatar for user Joshua Hu
    What is a vector
    (0 votes)
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    • duskpin ultimate style avatar for user Tejas Pandey
      A vector is a quantity or phenomenon that has two independent properties: magnitude and direction. The term also denotes the mathematical or geometrical representation of such a quantity.

      Examples of vectors in nature are velocity, momentum, force, electromagnetic fields, and weight. (Weight is the force produced by the acceleration of gravity acting on a mass.) A quantity or phenomenon that exhibits magnitude only, with no specific direction, is called a scalar . Examples of scalars include speed, mass, electrical resistance, and hard-drive storage capacity.
      (69 votes)
  • aqualine ultimate style avatar for user Mena S
    People say i am to young for liner algebra, I am in 6th grade is that true?
    (23 votes)
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  • blobby green style avatar for user PlastiiicToy
    It would be amazing if you added quizzes to the entirety of linear algebra course. In the other courses this helped me a tremendous amount and the lack of the quizzes here are keeping me from retaining the information. Thank you a ton, Khan academy!
    (40 votes)
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  • male robot donald style avatar for user Maik Lomeu
    When i build a matrix of my vector in a 2 dimensions plan, do I need to build it 2 in rows and 1 column or it works on a 1 row and 2 columns matrix? Sorry for bad english.. Thanks
    (10 votes)
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    • leaf green style avatar for user Lucas Van Meter
      When we start using matrices we write vectors as columns. So a 2-dimensinoal vector would have 2 rows and 1 column. This can be confusing because people will often write a vector as a string of number like (1,2) but they really mean a column vector. It should make more sense when you start using matrices to transform vectors into other vectors.
      (23 votes)
  • starky sapling style avatar for user Nightmare252
    How would you draw out a vector of more than 3 dimensions?
    (8 votes)
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    • leaf blue style avatar for user Gabriel Fonseca
      You wouldn't - We simply can't "draw" a vector of more than 3 dimensions. What we can do, however, is use some clever tricks to represent the fourth spatial dimension as something else. One could, for instance, make a small movie, where every second elapsed is equal to moving one unit on the fourth dimension. Another trick is to colour-code the vector, trying to represent it's coordinates in the fourth axis as a change in it's colours.

      So, in essence, you can't really "draw" a 4D vector, but we can get clever in how to convey it's meaning.
      (17 votes)
  • aqualine tree style avatar for user Jordan Eddy
    I'm confused about why you can have a speed as a magnitude. Shouldn't a magnitude be a distance instead of a speed? Aren't cartesian coordinate systems supposed to convey placement instead of some obscure km/h / mph value?
    (4 votes)
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    • aqualine ultimate style avatar for user Kyler Kathan
      Vectors can represent anything. Usually they represent position in an x, y, and z coordinate, but they're often used to show velocity in the x, y, and z direction or even acceleration in those directions. Many 2D graphics programs use an R4 vector system with matrices to represent an image's position, skew, rotation, etc.. Magnitude is nothing more than a number, so a vector with a magnitude of 5 could mean that something's 5 units from the point of origin, or that it's moving at 5 units per second or that there's 5 particles passing through a point per second. There's really no limit to the things you can represent with vectors.
      (19 votes)
  • spunky sam red style avatar for user Marvin Cohen
    How is the subject "linear algebra" different from "abstract algebra"?
    (5 votes)
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    • leafers seedling style avatar for user Phillis Alapatt
      Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields. However, Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
      (13 votes)
  • starky sapling style avatar for user dominic
    R^2 or R^3 means that it is the real number space multiplied by itself twice or three times? I am having a hard time seeing if that notation is supposed to be exponential or not...
    (5 votes)
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    • primosaur ultimate style avatar for user Derek M.
      Yes it is exponentiation, but not in the "standard" notion of multiplication. The multiplication done here is an operation on sets, the operation being called the "Cartesian Product". If A and B are two sets, then we denote their Cartesian product as AxB, or BxA depending on the order. Note the order does matter, as AxB is in general not equal to BxA. The set AxB is the set of ordered pairs (x, y), where x is in A, and y is in B.

      So RxR=R^2 is the collection of ordered pairs (x, y), where both x and y are real numbers. Likewise, the double Cartesian Product (RxR)xR=Rx(RxR)=R^3 is the collection of ordered triples (x, y, z), where x, y, and z are all real numbers.
      (9 votes)
  • mr pants teal style avatar for user Moon Bears
    Can anyone recommend a good linear algebra book? It'll be my second time seeing Linear Algebra.
    (5 votes)
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    • leaf red style avatar for user Bob Fred
      if you're already familiar with linear algebra, i would recommend looking through linear algebra again in a more theoretical way, AKA with vector spaces and axioms. the proofy version of linear algebra. this is, dare i say it, the right way to learn linear algebra.
      (6 votes)
  • blobby green style avatar for user Wilgen
    How can you use the Pythagorean theorem with vectors? What does using the Pythagorean theorem help you to find out about a vector?
    (5 votes)
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Video transcript

A vector is something that has both magnitude and direction. Magnitude and direction. So let's think of an example of what wouldn't and what would be a vector. So if someone tells you that something is moving at 5 miles per hour, this information by itself is not a vector quantity. It's only specifying a magnitude. We don't know what direction this thing is moving 5 miles per hour in. So this right over here, which is often referred to as a speed, is not a vector quantity just by itself. This is considered to be a scalar quantity. If we want it to be a vector, we would also have to specify the direction. So for example, someone might say it's moving 5 miles per hour east. So let's say it's moving 5 miles per hour due east. So now this combined 5 miles per are due east, this is a vector quantity. And now we wouldn't call it speed anymore. We would call it velocity. So velocity is a vector. We're specifying the magnitude, 5 miles per hour, and the direction east. But how can we actually visualize this? So let's say we're operating in two dimensions. And what's neat about linear algebra is obviously a lot of what applies in two dimensions will extend to three. And then even four, five, six, as made dimensions as we want. Our brains have trouble visualizing beyond three. But what's neat is we can mathematically deal with beyond three using linear algebra. And we'll see that in future videos. But let's just go back to our straight traditional two-dimensional vector right over here. So one way we could represent it, as an arrow that is 5 units long. We'll assume that each of our units here is miles per hour. And that's pointed to the right, where we'll say the right is east. So for example, I could start an arrow right over here. And I could make its length 5. The length of the arrow specifies the magnitude. So 1, 2, 3, 4, 5. And then the direction that the arrow is pointed in specifies it's direction. So this right over here could represent visually this vector. If we say that the horizontal axis is say east, or the positive horizontal direction is moving in the east, this would be west, that would be north, and then that would be south. Now, what's interesting about vectors is that we only care about the magnitude in the direction. We don't necessarily not care where we start, where we place it when we think about it visually like this. So for example, this would be the exact same vector, or be equivalent vector to this. This vector has the same length. So it has the same magnitude. It has a length of 5. And its direction is also due east. So these two vectors are equivalent. Now one thing that you might say is, well, that's fair enough. But how do we represent it with a little bit more mathematical notation? So we don't have to draw it every time. And we could start performing operations on it. Well, the typical way, one, if you want a variable to represent a vector, is usually a lowercase letter. If you're publishing a book, you can bold it. But when you're doing it in your notebook, you would typically put a little arrow on top of it. And there are several ways that you could do it. You could literally say, hey 5 miles per hour east. But that doesn't feel like you can really operate on that easily. The typical way is to specify, if you're in two dimensions, to specify two numbers that tell you how much is this vector moving in each of these dimensions? So for example, this one only moves in the horizontal dimension. And so we'll put our horizontal dimension first. So you might call this vector 5, 0. It's moving 5, positive 5 in the horizontal direction. And it's not moving at all in the vertical direction. And the notation might change. You might also see notation, and actually in the linear algebra context, it's more typical to write it as a column vector like this-- 5, 0. This once again, the first coordinate represents how much we're moving in the horizontal direction. And the second coordinate represents how much are we moving in the vertical direction. Now, this one isn't that interesting. You could have other vectors. You could have a vector that looks like this. Let's say it's moving 3 in the horizontal direction. And positive 4. So 1, 2, 3, 4 in the vertical direction. So it might look something like this. So this could be another vector right over here. Maybe we call this vector, vector a. And once again, I want to specify that is a vector. And you see here that if you were to break it down, in the horizontal direction, it's shifting three in the horizontal direction, and it's shifting positive four in the vertical direction. And we get that by literally thinking about how much we're moving up and how much we're moving to the right when we start at the end of the arrow and go to the front of it. So this vector might be specified as 3, 4. 3, 4. And you could use the Pythagorean theorem to figure out the actual length of this vector. And you'll see because this is a 3, 4, 5 triangle, that this actually has a magnitude of 5. And as we study more and more linear algebra, we're going to start extending these to multiple dimensions. Obviously we can visualize up to three dimensions. In four dimensions it becomes more abstract. And that's why this type of a notation is useful. Because it's very hard to draw a 4, 5, or 20 dimensional arrow like this.