Vectors - Magnitude & Direction

Maths: Edexcel A Level Pure Maths, topic 10.1.1

Vector components

A very useful way of writing a vector is in terms of its $x$ and $y$ -components.

Component form of a vector

Component form of a vector

In the $xy$ -coordinate system, a point on a plane is described by a pair of coordinates $(x, y)$ .
Similarly, a two dimensional vector, $\mathbf a$ is described by a pair of its vector coordinates:
$\mathbf{a} = a_i\mathbf{i} + a_j\mathbf{j}$
Where $\mathbf{i}$ and $\mathbf{j}$ are the unit vectors in the positive $x$ and $y$ directions respectively.

Column vector

Column vector

The equivalent vector written in column form is:
$\mathbf{a} = {a_i \choose a_j}$

$Notation$

Notation

Scalars are written as normal text, for example $a_i$
Vectors are often written in bold, underline or with an arrow on top, for example $\mathbf{a}\text{,}\; \underline{a}\;\text{or}\; \overrightarrow{a}$ .
When doing your working out, be sure to pick a notation and stick with it.
We suggest using the underline notation, as it is the quickest and easiest to write.

Magnitude

Magnitude

The magnitude of a vector written in component form is found by using Pythagoras' theorem:
$|\mathbf{a}| = \sqrt{{a_i}^2 + {a_j}^2}$
Where $|\mathbf{a}|$ is the magnitude of the vector and $a_i\text{,}\;a_{j}$ are the $x\text{ and }y$ -components of the vector.

Direction

Direction

The direction of the vector can be worked out by using trigonometry:
$\text{tan}\,\theta = \frac{\large a_j}{\large a_i}$
Where $\theta$ is the angle the vector $\mathbf{a}$ makes with the horizontal.

Test yourself on Vectors - Magnitude & Direction

Which of the following notations is NOT correct for a vector?

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