The set is defined as

Each element of is called a vector and consists of a sequence of real numbers. We call the -th coordinate of the vector .

For example, vectors in are of the form for some real numbers and . These vectors can be seen as points lying in a plane. In the vectors have three coordinates and can be visualized as points in (three dimensional) space.

Vector addition is an operation that takes two vectors and produces a third vector from them. In we define addition as follows; ,

This addition is called "coordinate-wise" addition because the -th coordinate of the sum of two vectors is the sum of their -th coordinates. That is gives .

Scalar multiplication is another operation on vectors which takes a real number and multiplies it against every coordinate of the vector. Take and then

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Formally, a vector space consists of a set of vectors , a field of scalars (normally or ), a zero vector , and two operations called vector addition,

and scalar multiplication,

These operations satisfy all of the following: for all and we have

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The dot product is an operation that takes two vectors in and produces a scalar.

The dot product satisfies the following properties for vectors and scalars ; Also, only when is the zero vector.

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The norm of a vector is its length. We define the Euclidean norm for vectors in as follows;

The norm satisfies for any vector and scalar . Also, the only vector with norm zero is the zero vector.

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Cauchy-Schwartz Inequality:

Triangle Inequality:

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The angle between two vectors can be defined as follows. Consider the plane which passes through and the origin. In this plane, you can draw a unit circle which is cut in two places by and . The length of the shortest arc of the cut circle gives the angle between and . To compute the angle between and , we use the following formula;

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Two vectors are orthogonal if the angle between them is . Equivalently, and are orthogonal if .

The argument is as follows. If and are orthogonal then the angle between them is . So, . Hence . Multiplying by gives . On the other hand, if then and hence .

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