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, matlabMathematica does not distinguish column vectors from row vectors, unless the user specifies which one is defined. One can define vectors using matlabMathematica commands: List, Table, Array, or curly brackets.

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

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 matlab 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 matlabMathematica as rows or columns, but rather by “size.”

In matlab, defining vectors and matrices is done by typing every row by inputing entries with or without comas:

The ones, zeros linspace, and logspace functions allow for explicit creations of vectors of a specific size and with a prescribed spacing between the elements. These functions will be demonstrated by example without providing an exhaustive reference. To create a vector with one of these functions you must (atleast initially) decide how long do you want the vector to be. You must also decide whether the vector is a row or column vector. The ones and zeros functions have two arguments. The first is the number of rows in the matrix you wish to create. The second is the number of columns. To create a row or a column vector set the appropriate argument of ones and zeros to one. To create a row vector of length 5, filled with ones use

You can define a vector with functions as compenets
Click to see matlab code

syms x
v = [1,2^6,sin(x)]

pkg load symbolic
syms x
v = [1,2^6,sin(x)]
    

So v is a vector with three components, v(1) =1, v(2)= 2^6, and v(3)=sin(x). Parenthesis notation is abbreviation for the matlab command 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

A column vector can be constructed from square brackets shown here [ ]. A semicolumn delineates each row. The output, however, may not look like a column vector.

Constructing a row vector is very similar to constructing a column vector, except two sets of semicolumns are eliminated.

Object Sizes

• ndims (a) Return the number of dimensions of a
• columns (a) Return the number of columns of a.
  Octave’s data structure type doesn’t have rows or columns, so the rows and columns functions return -1 for structure arguments.
• rows (a) Return the number of rows of a.
• numel (a) Return the number of elements in the object a.
• length (a) Return the length of the object a.
• size (a) Return a row vector with the size (number of elements) of each dimension for the object a.
• isempty (a) Return true if a is an empty matrix (any one of its dimensions is zero).
• : isnull (x) Return true if x is a special null matrix, string, or single quoted string.
• : sizeof (val) Return the size of val in bytes.
• squeeze (x) Remove singleton dimensions from x and return the result.
  Note that for compatibility with matlab, all objects have a minimum of two dimensions and row vectors are left unchanged.

function size_test

    nRepeats = 10000;

    times1 = zeros(nRepeats, 1);
    times2 = zeros(nRepeats, 1);
    times3 = zeros(nRepeats, 1);

    for k = 1:nRepeats
        data = rand(10000, 1);
        tic
        size(data, 1);
        times1(k) = toc;
        tic
        length(data);
        times2(k) = toc;
        tic
        numel(data);
        times3(k) = toc;
    end

    % Compute the total time required for each method
    fprintf('size:\t%0.8f\n', sum(times1));
    fprintf('length:\t%0.8f\n', sum(times2));
    fprintf('numel:\t%0.8f\n', sum(times3));
end

Example 1: In 3-dimensional space, the electric field E of a point charge q₁ with position vector r₁ at a point P---called the field point---with position vector r is given by the formula

\[ {\bf E} = \frac{k_e q_1}{\left\vert {\bf r} - {\bf r}_1 \right\vert^3} \left( {\bf r} - {\bf r}_1 \right) \]

We define the position vectors in matlabMathematica:

r = {x,y,z}; r1 ={x1,y1,z1} ;

When we place a semicolon at the end of an expression, matlabMathematica does not provide any output. Next, we will write an expression for the field (with k_e = 1). Recall that in matlabMathematica vector names are followed by underscores when being called in a function.

EField[r_ , r1_ , q1_ ] := q1/((r-r1).(r-r1))^(3/2) (r-r1)

We now take only the first two components of the field and try to make a two-dimensional plot of the field lines

{E1x,E1y} = Take[EField[{x,y,0},{1,1,0},1],2];

where Take[list, n] takes the first n elements of list and make a new list of them.

VectorPlot[{E1x, E1y}, {x, 0, 2}, {y, 0, 2}, Axes -> True]

Now we introduce another location of the second charge

r2 = {x2,y2,z2} ;

Then calculate the field due to this charge at the same field point

EField2[r_ , r2_ , q2_ ] := q2/((r-r2).(r-r2))^(3/2) (r-r2)

Now add the two fields to get the total electric field at r:

Etotal[r_, r1_, r2_, q1_, q2_] = EField[r,r1,q1] + EField2[r , r2 , q2 ]

Let us see what the field lines of a dipole look like. A dipole is a combination of two charges of equal strength and opposite signs. Let the positive charge of +1 be at (1,1,0) and the negative charge of -1 be at (2,1,0). Let us also assume that the field point is at (x,y,0). Since we are interested in a two-dimensional plot of the field lines, we separate the first two components of Etotal

{Etotal1, Etotal2} = vEtotal[r_] = EField[r, {1, 1, 0}, 1] + EField2[r, {2, 1, 0}, -1]
Take[Etotal[{x, y, 0}, 2];

Now we ready to plot this field

StreamPlot[{Etotal1, Etotal2}, {x, 0, 3}, {y, 0, 2}, Axes -> True, Ticks -> None]

		Another version of the same plot: charge[q_, {x0_, y0_, z0_}][x_, y_, z_] := q/((x - x0)^2 + (y - y0)^2 + (z - z0)^2)^(3/2) {x - x0, y - y0, z - z0}; projector[{x_, y_, z_}] := {x, y} VectorPlot[ projector[ charge[1, {0, 4, 0}][x, y, 0] + charge[-1, {0, -4, 0}][x, y, 0]], {x, -10, 10}, {y, -10, 10}]; StreamPlot[ projector[ charge[1, {-2, 0, 0}][x, y, 0] + charge[-1, {2, 0, 0}][x, y, 0]], {x, -5, 5}, {y, -5, 5}]
Electric field potential of a dipole.		matlabMathematica code

First, we define the Coulomb fields at the origin using "Ec" in the code below

Ec[x_,y_] := {x/(x^2 + y^2)^(3/2), y/(x^2 + y^2)^(3/2)};

		A complitely different vector field is obtained when we add two equal charges: Ec[x_, y_] := {x/(x^2 + y^2)^(3/2), y/(x^2 + y^2)^(3/2)}; StreamPlot[Ec[x + 2, y] + Ec[x - 2, y], {x, -5, 5}, {y, -5, 5}]
Electric field potential of two equal charges.		Mathematica code

We can visualize vector fields in 3-dimensional space.    ■

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.

matlab has three multiplication commands for vectors: the dot (or inner), 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 matlabMathematica 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 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

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 \( {\bf v}^{\ast} = \overline{{\bf v}^{\mathrm T}} . \)

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 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.