# Section 1.1.1: Vectors

If you found vectors confusing, don't worry, this explanation is intended to clarify what is really a very simple idea.

A vector may be thought of simply as a list of `things'. They may be any kind of things. The list may contain things, or elements, which are obviously similar, or apparently different. There will normally be some kind of similarity to justify their inclusion in the same list.

The most common reason in our case for listing elements together is that they are all quantities which form the solution to a set of equations. The elements of the vector then represent variables or symbols which are unknowns in a mixed set of equations. Physically, the elements may represent flows, temperatures, pressures, compositions, costs or indeed anything.

The reason for putting these disparate quantities into the same list is that of abstraction. The entire list may be represented by a single symbol and in computer terms, by a single construct in the computer. It may then be manipulated mathematically or computationally by a single operation or successive set of operations. We need not always be aware of the details of these operations if a mathematical function or computer program has already been provided for us to use, as will often be the case.

### Notation

The list of a vector is usually written in square or curly brackets, e.g.:

{a, b, c, d} or [a, b, c, d]
for a vector containing elements a, b, c and d.

(Note that vector elements, like other symbols used in mathematical equations, are usually set in a slanted font to distinguish them from ordinary text.)

It may be convenient to consider the list to be arranged either horizontally as above or vertically, e.g.:

Conventionally, vectors written in square brackets are thought of as being arranged as written, while a horizontal list in curly brackets (as in the first example) is interpreted as a space-saving way of writing a vertical list. We will usually adopt this convention.

A single symbol standing for a complete list or vector is usually set in boldface type or underlined. They may be set in either slanted or straight type. We will normally use boldface straight type in these notes, e.g.:

l = { a, b, c, d}

If the names of the elements are in lower case then the name of the vector should normally also be lower case. It will not always be convenient or possible to adhere to these conventions however.

### Examples

In a simple chemical process there are three streams of material. Let these be s1, s2 and s3. We can refer to the complete list of streams in the process as s where:

s = { s1, s2, s3}

In fact each stream may be though of as having a set of properties, i.e. temperature, pressure and the flowrates of the species present in the process. Each stream may also written as a vector, e.g.:

s1 = {T1, P1, f1}

Here T and P are temperatures and pressures, and f1 is yet another list or vector of the flows of all the species in the stream.

It will thus be seen that vectors can contain other vectors and may, if desired be given subscripts to distinguish them. Elements of one vector may often also be considered to be members of other lists, e.g. to refer to all the pressures in the process, we may define:

P = {P1, P2, P3}

### Special cases

Certain special cases of vectors are used in particular ways, but these are all specialisations of the general idea of the vector as a list. For example:
• A vector velocity is a 2 element list containing a magnitude and a direction, although it is seldom explicitly written in that way.
• Geometric vectors consist of a list of typically three coordinates x, y and z for position in space.
• In electrical engineering, and in some aspects of process control, an alternating current is thought of as a two element vector or list of amplitude and phase.
• Complex numbers, having a real and imaginary part, are obviously two element vectors. Since mathematical operations are defined for complex numbers, this provides a way of defining mathematical operations on two element vectors. That is why complex numbers are used in electrical engineering and to represent fluid flow in two dimensions.

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