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1.1 Definition and algebraic representation

Vectors are mathematical concepts that express not just magnitudes, but magnitudes with directions.

Example 1.1

The speed of a car does not depend on the direction it is travelling in, but its velocity does. The driver who was arrested for a “velocity offence” was travelling the wrong way down the motorway!

Some direction-less physical properties: speed, density, energy, mass.

Some physical properties which do have directions: force, acceleration, velocity.

For example, the earth’s gravitational force acts in a specific direction: it pulls us downwards (or more precisely, towards the centre of mass of the earth).

A vector is a directed line segment, written $\overrightarrow{AB}$ where $A$ is the start-point (or tail) and $B$ is the end-point (or head). The length of $A B$ gives the magnitude of the vector. A parallel line segment $A^{\prime}B^{\prime}$ with the same length, orientation and
direction represents the same vector. In other words, the
starting point of the line segment is not part of the
specification of the vector.

Vectors in ${\mathbb{R}}^{2}$

Given a vector in the plane, we may choose as its representative the directed line segment that has its tail at the origin. This directed line segment is uniquely determined by the coordinates of its head, which in ${\mathbb{R}}^{2}$ can be given by a pair of coordinates $x$ , $y$ . So there is a one-to-one correspondence between points and vectors in the plane. We will write points in ${\mathbb{R}}^{2}$ with a comma: $(x,y)$ . The same notation may be used for vectors in ${\mathbb{R}}^{2}$ . However, if we want to clearly distinguish points from vectors, then we may also write vectors with a space: $(x\;\;y)$ . (Note that this could more precisely be termed a row vector, as opposed to the column vector $\left(\begin{array}[]{cc}x\\ y\end{array}\right)$ .)

Example 1.2

Some revision: in each of the following cases write the vector $\overrightarrow{AB}$ in the form $(x\;\;y)$ :

i) $A=(1,-2)$ , $B=(-3,2)$ , $\;\;\;$ ii) $A=(-2,7)$ , $B=(2,6)$ , $\;\;\;$ iii) $A=(-1,1)$ , $B=(-1,4)$ .

Vectors in ${\mathbb{R}}^{3}$ , or in ${\mathbb{R}}^{n}$

Similar principles operate for vectors in ${\mathbb{R}}^{n}$ , for $n\geq 3$ . The easiest case to visualize is ${\mathbb{R}}^{3}$ , which corresponds to the three physical dimensions we see around us. The points in ${\mathbb{R}}^{3}$ are described by three coordinate values, usually labelled $x$ , $y$ and $z$ . The axes for $x, y, z$ should be in the respective directions of the thumb, first and second fingers on your right hand.

Often (though not always) we will think of the $z$ -axis as coming out of the page, so that a point in space which is below the paper will have a negative value of $z$ , and a point which is above the page will have a positive value of $z$ . We will usually write points in ${\mathbb{R}}^{3}$ in the form $(x,y,z)$ .

Once more we define a vector to be a directed line segment, and we do not include the start-point as part of the definition. Analogous to the $2$ -dimensional case, a vector in ${\mathbb{R}}^{3}$ is written $(x,y,z)$ or $(x\;\;y\;\;z)$ .

Example 1.3

The vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are the same, where $A=(1,2,3)$ , $B=(4,5,6)$ and $C=(1,1,1)$ , $D=(4,4,4)$ because:

In this course we will almost always deal with vectors in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$ , but you should be aware that similar principles apply to ${\mathbb{R}}^{n}$ for any $n$ , even $n=1$ or $n=0$ !