Home page for accesible maths 1 Vectors 1.1 Definition and algebraic representation 1.3 Addition of vectors

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1.2 Scalar multiplication and unit vectors

What is the length of the vector $(x\;y)$ ? It is the distance from the start point $(0,0)$ to the end-point $(x,y)$ . By Pythagoras’ theorem, this is $\sqrt{x^{2}+y^{2}}$ . Similarly, the length of the vector $(x\;\;y\;\;z)$ is $\sqrt{x^{2}+y^{2}+z^{2}}$ (by repeated application of Pythagoras’ theorem). In general, we have the obvious definition:

Definition 1.4

The modulus (or magnitude) of a vector ${\bf a}=(x_{1}\;\;\ldots\;\;x_{n})\in{\mathbb{R}}^{n}$ , denoted $|{\bf a}|$ , is its length: $\sqrt{x_{1}^{2}+\ldots+x_{n}^{2}}$ . A unit vector is a vector ${\bf a}$ such that $|{\bf a}|=1$ .

For example, if ${\bf u}=(1\;\;2\;\;-1\;\;0\;\;3)\in{\mathbb{R}}^{5}$ , then $|{\bf u}|=$

Multiplication by scalars

One important operation on vectors is scalar multiplication.

{\bf a}=(x_{1}\;\;\ldots\;\;x_{n})

and

\lambda

is a real number, we define

\lambda{\bf a}=(\lambda x_{1}\;\;\ldots\;\;\lambda x_{n}).

We write

-{\bf a}

for

(-1){\bf a}

Some physical interpretations

If a car doubles its speed, but keeps moving in (exactly) the same direction, then its velocity has gone from ${\bf v}$ to $2{\bf v}$ .

If the position of person $B$ with respect to person $A$ is expressed as the vector ${\bf v}=\overrightarrow{AB}$ , then the position of $A$ with respect to $B$ is expressed as $-{\bf v}=\overrightarrow{BA}$ .

Lemma 1.5

Let ${\bf a}$ be a vector in ${\mathbb{R}}^{n}$ and let $\lambda\in{\mathbb{R}}$ . We have: $|\lambda{\bf a}|=|\lambda|.|{\bf a}|$ .

Proof

$\square$

Lemma 1.6

Let ${\bf a}$ be a non-zero vector in ${\mathbb{R}}^{n}$ . Then the unit vector in the direction of ${\bf a}$ is $\frac{1}{|{\bf a}|}{\bf a}$ .

The unit vector in the opposite direction to ${\bf a}$ is $-\frac{1}{|{\bf a}|}{\bf a}$ .

Proof

$\square$

Easy but important examples: Find the unit vector in the direction of:

(i) $(2\;4)$ , $\;\;$ (ii) $(3\;4)$ , $\;\;$ (iii) $(1\;\;-1\;\;4)$ .

Unit vectors in ${\mathbb{R}}^{2}$

There is a straightforward way to describe all unit vectors in ${\mathbb{R}}^{2}$ , using the angle to the $x$ -axis. Let ${\bf u}$ be a unit vector in ${\mathbb{R}}^{2}$ which is at an angle $\theta$ to the $x$ -axis, where $0\leq\theta<2\pi$ . Then we can use elementary trigonometry to determine the $x$ - and $y$ -coordinates of ${\bf u}$ :

Hence we see that ${\bf u}=(\cos\theta\;\;\sin\theta)$ . More generally, if $|{\bf u}|=r>0$ and ${\bf u}$ makes an angle $\theta$ with the $x$ -axis, then ${\bf u}=(r\cos\theta\;\;r\sin\theta)$ . The values $r,\theta$ are the polar coordinates of ${\bf u}$ . We will see more about these later in the course.