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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.

## Want to join the conversation?

• What is a vector •   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.
• People say i am to young for liner algebra, I am in 6th grade is that true? • 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! • 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 • 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.
• How would you draw out a vector of more than 3 dimensions? • 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.
• 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? • 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.
• How is the subject "linear algebra" different from "abstract algebra"? • 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.
• 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... • 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.  