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Vector is a basic [mathematical](math.md) object that expresses direction and magnitude (such as velocity, force etc.) and is very often expressed as (and many times misleadingly equated with) an "[array](array.md) of [numbers](number.md)". Nevertheless in [programming](programming.md) one dimensional arrays are somewhat synonymous with vectors -- for example in two dimensional space an array `[4,3]` expresses a vector pointing 4 units to the "right" (along X axis) and 3 units "up" (along Y axis) and has the magnitude 5 (which is the vector's [length](length.md)). Vectors are one of the very basic concepts of advanced math and are used almost in any advanced area of math, physics, programming etc. -- basically all of physics and engineering operates with vectors, programmers will mostly encounter them in areas such as computer [graphics](graphics.md) (e.g 3D graphics, 2D [vector graphics](vector_graphic.md), ...), [physics engines](physics_engine.md) (forces, velocities, acceleration, ...), [machine learning](machine_learning.md) (feature vectors, ...) or [signal processing](signals.md) (e.g. [Fourier transform](fourier_transform.md) just interprets a signal as a vector and transforms it to a different basis) etc. In this article we will implicitly focus on vectors from programmer's point of view (i.e. "arrays of numbers") which to a mathematician will seem very simplified, but we'll briefly also foreshadow the mathematical view.
(NOTE: the term *vector* is used a lot in different contexts and fields, usually with some connection to the mathematical idea of vector which is sometimes however very loose, e.g. in low-level programming *vector* means a memory holding an address of some event handler. It's better to just look up what "vector" means in your specific area of interest.)
(NOTE: the term *vector* is used a lot in different contexts and fields, usually with some connection to the mathematical idea of vector which is sometimes however very loose, e.g. in low-level programming *vector* means a memory holding an address of some event handler (basically a pointer), probably because it "points" somewhere. It's better to just look up what "vector" means in your specific area of interest.)
Just like in elemental mathematics we deal with "simple" numbers such as 10, -2/3 or [pi](pi.md) -- we retrospectively call such "simple" numbers **[scalars](scalar.md)** -- advanced mathematics generalizes the concept of such a number into vectors ("arrays of numbers", e.g. `[1,0,-3/5]` or `[0.5,0.5]`) and yet further to [matrices](matrix.md) ("two dimensional arrays of numbers") and defines a way to deal with such generalizations into **[linear algebra](linear_algebra.md)**, i.e. we have ways to add and multiply vectors and matrices and solve [equations](equation.md) with them, just like we did in elemental algebra (of course, linear algebra is a bit more complex as it mixes together scalars, vectors and matrices). In yet more advanced mathematics the concepts of vectors and matrices are further generalized to **[tensors](tensor.md)** which may also be seen as "N dimensional arrays of numbers" but further add new rules and interpretation of such "arrays" -- vectors can therefore be also seen as a tensor (of rank 1) -- note that in this context there is e.g. a fundamental distinction between row and column vectors. Keep in mind that vectors, matrices and tensors aren't the only possible generalization of numbers, another one is e.g. that of [complex numbers](complex_number.md), [quaternions](quaternion.md), [p-adic numbers](p_adic.md) etc. Anyway, in this article we won't be discussing tensors or any of the more advanced concepts further, they are pretty non-trivial and mostly beyond the scope of mere programmer's needs :) We'll keep it at linear algebra level.
Just like in elemental mathematics we deal with "simple" numbers such as 10, -2/3 or [pi](pi.md) -- we retrospectively call such "simple" numbers **[scalars](scalar.md)** -- advanced mathematics generalizes the concept of such a number into vector ("arrays of numbers", e.g. `[1,0,-3/5]` or `[0.5,0.5]`) and yet further to [matrices](matrix.md) ("two dimensional arrays of numbers") and defines a way to deal with such generalizations into **[linear algebra](linear_algebra.md)**, i.e. we have ways to add and multiply vectors and matrices and solve [equations](equation.md) with them, just like we did in elemental algebra (of course, linear algebra is a bit more complex as it mixes together scalars, vectors and matrices). In yet more advanced mathematics the concepts of vectors and matrices are further generalized to **[tensors](tensor.md)** which may also be seen as "N dimensional arrays of numbers" but further add new rules and interpretation of such "arrays" -- vectors can therefore be also seen as a tensor (of rank 1) -- note that in this context there is e.g. a fundamental distinction between row and column vectors. Keep in mind that vectors, matrices and tensors aren't the only possible generalization of numbers, another one is e.g. that of [complex numbers](complex_number.md), [quaternions](quaternion.md), [p-adic numbers](p_adic.md) etc. Anyway, in this article we won't be discussing tensors or any of the more advanced concepts further, they are pretty non-trivial and mostly beyond the scope of mere programmer's needs :) We'll keep it at linear algebra level.
**Vector is not merely a coordinate**, though the traditional representation of it suggest such representation and programmers often use vector data types to store coordinates out of convenience (e.g. in 3D graphics engines vectors are used to specify coordinates of 3D objects); vector should properly be seen as a **direction and magnitude** which has **no position**, i.e. a way to correctly imagine a vector is something like an **arrow** -- for example if a vector represents velocity of an object, the direction (where the arrow points) says in which direction the object is moving and the magnitude (the arrow length) says how fast it is moving (its [speed](speed.md)), but it doesn't say the position of the object (the arrow itself records no position, it just "hangs in thin air").
**Vector is not merely a coordinate**, though the traditional representation of it suggest such representation and programmers often use vector data types to store coordinates out of convenience (e.g. in [3D graphics](3d_rendering.md) engines vectors are used to specify coordinates of 3D objects); vector should properly be seen as a **direction and magnitude** which has **no position**, i.e. a way to correctly imagine a vector is something like an **arrow** -- for example if a vector represents velocity of an object, the direction (where the arrow points) says in which direction the object is moving and the magnitude (the arrow length) says how fast it is moving (its [speed](speed.md)), but it doesn't say the position of the object (the arrow itself records no position, it just "hangs in thin air").
Watch out, **mathematicians dislike defining vectors as arrays of numbers** because vectors are essentially NOT arrays of numbers, such arrays are just one way to express them. Similarly we don't have to interpret any array of numbers as a vector, just as we don't have to interpret any string of letter as a word in human language. A vector is simply a direction and magnitude, an "arrow in space" of *N* dimensions; a natural way of expressing such arrow is through multiples of basis vectors (so called components), BUT the specific numbers (components) depend on the choice of basis vectors, i.e. the SAME vector may be written as an array of different numbers (components) in a different basis, just as the same concept of a [dog](dog.md) is expressed by different words in different languages. Even with the same basis vectors the numbers (components) depend on the method of measurement -- instead of expressing the vector as a linear combination of the *N* basis vectors we may express it as *N* [dot products](dot_product.md) with the basis vectors -- the numbers (components) will be different, but the expressed vector will be the same. Mathematicians usually define vectors abstractly simply as members of a **[vector space](vector_space.md)** which is a set of elements (vectors) along with operations of addition and multiplication which satisfy certain given rules ([axioms](axiom.md)).