6.3 Vectors and Coordinate Systems: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
Mathematics Focus
Vectors that have length [latex]1[/latex] are called unit vectors. See Language Focus 3 in Section 1.6 to learn more about '1-ness'. Unit vectors are important because they isolate and represent the direction of the vector, independently of the magnitude. Due to this useful property, we distinguish unit vectors by using the notation
The verb normalise and its noun form normalisation are used in mathematics with a precise technical meaning that is different from their everyday English meaning.
In everyday English, normalise means to make something normal, usual, or acceptable. For example: Society is gradually normalising remote working. The noun normalisation in this general sense refers to the process of making something conform to a standard or become widely accepted.
In mathematics, however, the meaning is more specific: to normalise a vector means to divide it by its magnitude, producing a unit vector that points in the same direction.
For more examples of verb-noun pairs see Language Focus in Section 6.5.
where the coefficients [latex]a_{1}, a_{2}, a_{3}[/latex] are called the components of vector [latex]{\bf a}[/latex].
This is read out as the vector a is equal to, a one i hat, plus a two j hat, plus a three k hat, which is equal to a one, comma a two, comma a three.
Mathematics Focus
We refer above to the 'three-dimensional Euclidean space' (often denoted as [latex]\mathbb{R}^3[/latex]). The term Euclidean refers to the geometry established by the ancient Greek mathematician Euclid.
Some properties of Euclidean spaces:
- Space is flat: The shortest path between two points is a straight line.
- Parallel lines never meet: If you have two parallel lines, they will stay the same distance apart forever.
- Angles in a triangle add to 180°: This is a key property that distinguishes Euclidean geometry from non-Euclidean geometries (like spherical geometry, where angles add up to more than 180°).
Dimensions:
- 1D Space ([latex]\mathbb{R}[/latex]): An infinite straight line, where one can only move backwards and forwards. This is often called the Euclidean line.
- 2D Space ([latex]\mathbb{R}^2[/latex]): An infinite flat plane (think of a sheet of paper), where one can move left-right and up-down. This is often called the Euclidean plane.
- 3D Space ([latex]\mathbb{R}^3[/latex]): This models the physical space around us. We add a third dimension (depth), allowing movement in three independent directions [latex](x, y, z)[/latex].
- n-dimensional space ([latex]\mathbb{R}^n[/latex]): In mathematics is often useful to consider the n-dimensional space, where [latex]n[/latex] is any positive integer.
Notation:
We write coordinates in [latex]\mathbb{R}^3[/latex] as an ordered triple [latex](x, y, z)[/latex] (see Language Focus 3 in Section 1.6 to learn more about '3-ness'). The symbol [latex]\mathbb{R}[/latex] is the set of Real numbers, and the power [latex]3[/latex] tells us there are three dimensions.
In terms of the individual components, the length/magnitude/modulus of vector [latex]{\bf a}[/latex] is given by
\[|{\bf a}|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}.\]
This is read out as the length/magnitude/modulus of a is equal to the square root of, a one squared plus, a two squared plus, a three squared.
\[{\bf a} \cdot {\bf b}=a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}.\]
This is read out as a dot b is equal to a one b one plus, a two b two plus, a three b three.
Given two vectors [latex]{\bf a}[/latex] and [latex]{\bf b}[/latex], their cross product (or vector product) is defined to be the vector quantity
\[{\bf a} \times {\bf b}=(a_{2} b_{3}-a_{3} b_{2}) \,{\bf\hat{i}}+(a_{3} b_{1}-a_{1} b_{3})\, {\bf\hat{j}}+(a_{1} b_{2}-a_{2} b_{1})\, {\bf\hat{k}}.\]
This is read out as a cross b is equal to, a two b three minus a three b two, i hat, plus, a three b one minus a one b three, j hat, plus, a one b two minus a two b one, k hat.
Language Focus
Observation 1: Note that the terms dot product and cross product are named after the symbols used in their mathematical notation: the dot ([latex]{\bf a} \cdot {\bf b}[/latex]) and the cross ([latex]{\bf a} \times {\bf b}[/latex]).
Observation 2: In arithmetic, there is only one way to multiply two numbers. However, in vector algebra, there are multiple ways to multiply vectors, and each operation has a specific meaning and result. Therefore, we must use precise terminology to distinguish them.
Types of multiplication:
- scalar multiplication: multiplying a vector by a scalar, e.g. [latex]\lambda{\bf a}[/latex].
- scalar product (or dot product): multiplying two vectors to produce a scalar quantity, e.g. [latex]{\bf a} \cdot {\bf b}[/latex].
- vector product (or cross product): multiplying two vectors to produce a vector quantity, e.g. [latex]{\bf a} \times {\bf b}[/latex].
- Scalar triple product: multiplying three vectors to produce a scalar quantity, e.g. [latex]{\bf a} \cdot ({\bf b} \times {\bf c})[/latex].
- Vector triple product: multiplying three vectors together to produce a vector quantity, e.g. [latex]{\bf a} \times ({\bf b} \times {\bf c})[/latex].
- The system using [latex](x, y)[/latex] is called the Cartesian coordinate system (named after mathematician René Descartes). Any coordinate system defined by two perpendicular number lines intersecting at an origin [latex](0,0)[/latex], (not necessarily the x-axis and y-axis) are called a rectangular coordinate systems. Note that the plural of axis is axes.
- The system using [latex](r, \theta)[/latex] is called the polar coordinate system (see Section 5.3 for more details and the analogy in complex numbers).
In three dimensions (3D), we often use:
- Cartesian coordinates extended to [latex](x, y, z)[/latex].
- Cylindrical coordinates [latex](r, \theta, z)[/latex], which are the extension of polar coordinates. Cylindrical coordinates are named as such, because they form cylinders.
- Spherical coordinates [latex](r, \theta, \phi)[/latex], which are called as such, because they form spheres.
Coordinates are read out in a similar way. For example, [latex](x, y, z)[/latex] is read out as x comma y comma z.
A right-handed coordinate system is one that follows the right-hand rule.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
