There are no universal names for the coordinates in the three axes. However, the horizontal axis is traditionally called abscissa borrowed from New Latin (short for linear abscissa, literally, "cut-off line"), and usually denoted by x. The next axis is called ordinate, which came from New Latin (linea), literally, line applied in an orderly manner; we will usually label it by y. The last axis is called applicate and usually denoted by z. Correspondingly, the unit vectors are denoted by i (abscissa), j (ordinate), and k (applicate), called the basis. Once rectangular coordinates are set up, any vector can be expanded through these unit vectors. In the three dimensional case, every vector can be expanded as \( {\bf v} = v_1 {\bf i} + v_2 {\bf j} + v_3 {\bf k} ,\) where \( v_1, v_2 , v_3 \) are called the coordinates of the vector v. Coordinates are always specified relative to an ordered basis. When a basis has been chosen, a vector can be expanded with respect to the basis vectors and it can be identified with an ordered n-tuple of n real (or complex) numbers or coordinates. The set of all real (or complex) ordered numbers is denoted by ℝⁿ (or ℂⁿ). In general, a vector in infinite dimensional space is identified by an infinite sequence of numbers. Finite dimensional coordinate vectors can be represented by either a column vector (which is usually the case) or a row vector. We will denote column-vectors by lower case letters in bold font, and row-vectors by lower case letters with a superimposed arrow. Because of the way the Wolfram Language uses lists to represent vectors, Mathematica does not distinguish column vectors from row vectors, unless the user specifies which one is defined. One can define vectors using Mathematica commands: List, Table, Array, or curly brackets.

In mathematics and applications, it is a custom to distinguish column vectors

for which we use lowercase letters in boldface type, from row vectors (ordered n-tuple) Here entries \( v_i \) are known as the component of the vector. The column vectors and the row vectors can be defined using matrix command as an example of an \( n\times 1 \) matrix and \( 1\times n \) matrix, respectively.

The concept of a vector space (also a linear space) has been defined abstractly in mathematics. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century; however, the idea crystallized with the work of the German mathematician Hermann Günther Grassmann (1809--1877), who published a paper in 1862. A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars, the result producing more vectors in this collection. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally scalars in any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms (they can be found on the web page).

Vectors in Mathematica are built, manipulated and accessed similarly to matrices (see next section). However, as simple lists (“one-dimensional,” not “two-dimensional” such as matrices that look more tabular), they are easier to construct and manipulate. They will be enclosed in brackets ( [,] ) which allows us to distinguish a vector from a matrix with just one row, if we look carefully. The number of “slots” in a vector is not referred to in Mathematica as rows or columns, but rather by “size.”

In Mathematica, defining vectors and matrices is done by typing every row in curly brackets:

So v is a vector with three components, v[[1]] =1, v[[2]]= 2^6, and v[[3]]=Sin[x]. Double bracket notation is abbreviation for the Mathematica command Part. Its entries can be numbers or functions or even vectors and other entities. We usually denote vectors with lower case letters while matrices with upper case letters. Say we define a \( 2\times 3 \) matrix (with two rows and three columns) as However, to see the traditional form of the matrix on the screen, one needs to add a corresponding command, either TraditionalForm or MatrixForm. These special commands tell Mathematica that the output should be displayed with the elements of list arranged in a regular array:

A column vector can be constructed from curly brackets shown here { }. A comma delineates each row. The output, however, may not look like a column vector. To fix this you must call //MatrixForm on your variable representation of a row vector.

Constructing a row vector is very similar to constructing a column vector, except two sets of curly brackets are used. Again the output looks like a row vector and so //MatrixForm must be called to put the row vector in the format that you more familiar with:

Mathematica usually does not distinguish column-vectors from row-vectors. For instance, let us define two vectors: So we see that a is a column vector, which is a matrix of dimension \( 3 \times 1 ,\) while b is a row vector, which is a matrix of dimension \( 1 \times 3 .\) When we multiply the matrix A by vector a from left or right, Mathematica treats this vector either as a \( 3 \times 1 \) matrix or as a \( 1 \times 3 \) vector: However, when we apply the matrix A to the vector b, Mathematica will not accept multiplication from the right: The following command finds the length (number of components) of a vector: That is, a linear combination of vectors from S is a sum of scalar multiples of those vectors. Let S be a nonempty subset of V. Then the span of S in V is the set of all possible (finite) linear combinations of the vectors in S (including the zero vector). It is usually denoted by span(S). In other words, a linear span of a set of vectors in a vector space is the subspace of V equal to the intersection of all subspaces containing that set. Example 1: Electric Fields. Example 2: Linear combination of vectors

Both a vector and a matrix can be multiplied by a scalar; with the operation being *. Matrices and vectors can be added or subtracted only when their dimensions are the same.

 

Mathematica has three multiplication commands for vectors: the dot (or inner) and outer products (for arbitrary vectors), and the cross product (for three dimensional vectors).

For three dimensional vectors \( {\bf a} = a_1 \,{\bf i} + a_2 \,{\bf j} + a_3 \,{\bf k} = \left[ a_1 , a_2 , a_3 \right] \) and \( {\bf b} = b_1 \,{\bf i} + b_2 \,{\bf j} + b_3 \,{\bf k} = \left[ b_1 , b_2 , b_3 \right] \) , it is possible to define special multiplication, called the cross-product:

The cross product can be done on two vectors. It is important to note that the cross product is an operation that is only functional in three dimensions. The operation can be computed using the Cross[vector 1, vector 2] operation or by generating a cross product operator between two vectors by pressing [Esc] cross [Esc]. ([Esc] refers to the escape button)

The dot product of two vectors of the same size \( {\bf x} = \left[ x_1 , x_2 , \ldots , x_n \right] \) and \( {\bf y} = \left[ y_1 , y_2 , \ldots , y_n \right] \) (regardless of whether they are columns or rows because Mathematica does not distinguish rows from columns) is the number, denoted either by \( {\bf x} \cdot {\bf y} \) or \( \left\langle {\bf x} , {\bf y} \right\rangle ,\)

when entries are real, or

when entries are complex. Here \( \overline{\bf x} = \overline{a + {\bf j}\, b} = a - {\bf j}\,b \) is a complex conjugate of a complex number x = a + jb.

The dot product of any two vectors of the same dimension can be done with the dot operation given as Dot[vector 1, vector 2] or with use of a period “. “ .

With Euclidean norm ‖·‖₂, the dot product formula defines θ, the angle between two vectors. The dot product was first introduced by the American physicist and mathematician Josiah Willard Gibbs (1839--1903) in the 1880s. ■

An outer product is the tensor product of two coordinate vectors \( {\bf u} = \left[ u_1 , u_2 , \ldots , u_m \right] \) and \( {\bf v} = \left[ v_1 , v_2 , \ldots , v_n \right] , \) denoted \( {\bf u} \otimes {\bf v} , \) is an m-by-n matrix W of rank 1 such that its coordinates satisfy \( w_{i,j} = u_i v_j . \) The outer product \( {\bf u} \otimes {\bf v} , \) is equivalent to a matrix multiplication \( {\bf u} \, {\bf v}^{\ast} , \) (or \( {\bf u} \, {\bf v}^{\mathrm T} , \) if vectors are real) provided that u is represented as a column \( m \times 1 \) vector, and v as a column \( n \times 1 \) vector. Here \( {\bf v}^{\ast} = \overline{{\bf v}^{\mathrm T}} . \)

Example 3: Outer product

An inner product of two vectors of the same size, usually denoted by \( \left\langle {\bf x} , {\bf y} \right\rangle ,\) is a generalization of the dot product if it satisfies the following properties:

• \( \left\langle {\bf v}+{\bf u} , {\bf w} \right\rangle = \left\langle {\bf v} , {\bf w} \right\rangle + \left\langle {\bf u} , {\bf w} \right\rangle . \)
• \( \left\langle {\bf v} , \alpha {\bf u} \right\rangle = \alpha \left\langle {\bf v} , {\bf u} \right\rangle \) for any scalar α.
• \( \left\langle {\bf v} , {\bf u} \right\rangle = \overline{\left\langle {\bf u} , {\bf v} \right\rangle} , \) where overline means complex conjugate.
• \( \left\langle {\bf v} , {\bf v} \right\rangle \ge 0 , \) and equal if and only if \( {\bf v} = {\bf 0} . \)

The fourth condition in the list above is known as the positive-definite condition. A vector space together with the inner product is called an inner product space. Every inner product space is a metric space. The metric or norm is given by

The nonzero vectors u and v of the same size are orthogonal (or perpendicular) when their inner product is zero: \( \left\langle {\bf u} , {\bf v} \right\rangle = 0 . \) We abbreviate it as \( {\bf u} \perp {\bf v} . \) If A is an n × n positive definite matrix and u and v are n-vectors, then we can define the weighted Euclidean inner product In particular, if w₁, w₂, ... , w_n are positive real numbers, which are called weights, and if u = ( u₁, u₂, ... , u_n) and v = ( v₁, v₂, ... , v_n) are vectors in ℝⁿ, then the formula defines an inner product on \( \mathbb{R}^n , \) that is called the weighted Euclidean inner product with weights w₁, w₂, ... , w_n. Example 4: Weighted inner product. Example 5: Set of integrable functions Example 6: Set of polynomials

The invention of Cartesian coordinates in 1649 by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

 

In order to define how close two vectors are, and in order to define the convergence of sequences of vectors, we can use the notion of a norm. We will heavily use the following notation for nonnegative real numbers: