3.5. Vector magnitude#
The magnitude of a vector \(\vec{a}\) is the distance between the head and tail of \(\vec{a}\) which we can calculate using an extension of Pythagoras’ theorem.
Definition 3.4 (Vector magnitude)
The magnitude of a vector \(\vec{a} = (a_1, a_2, \ldots, a_n)^\mathsf{T}\) denoted by \(|\vec{a}|\) is calculated using
Vector magnitude is also known as the Euclidean norm. Note that \(|\vec{a}|=0\) if and only if \(\vec{a}=(0, 0, \ldots, 0)^\mathsf{T}\).
Example 3.2
Calculate the magnitudes of the following vectors
(i) \(\vec{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\);
Solution (click to show)
(ii) \(\vec{v} = \begin{pmatrix} 5 \\ -12 \\ 0 \end{pmatrix}\);
Solution (click to show)
(iii) \(\vec{w} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\).
Solution (click to show)
3.5.1. Unit vectors#
For every non-zero vector \(\vec{a}\) there exist a unique unit vector which is a vector in the same direction as \(\vec{a}\) and whose magnitude is 1.
Definition 3.5 (Unit vectors)
A unit vector is a vector with a magnitude of 1.
Theorem 3.2 (Normalising a vector)
Any non-zero vector can be scaled to transform it into a unit vector by dividing all its coordinates by its magnitude
This process is called normalising a vector. Unit vectors are denoted with a caret above the vector name, i.e., \(\hat{\vec{a}}\) which is read as ‘a hat’.
Proof. Let \(\vec{a}\) be a non-zero vector
Example 3.3
Find the unit vector parallel to the following:
(i) \(\vec{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\);
Solution (click to show)
Check magnitude of \(\hat{\vec{u}}\)
(ii) \(\vec{v} = \begin{pmatrix} 5 \\ -12 \\ 0 \end{pmatrix}\).
Solution (click to show)
Check magnitude of \(\hat{\vec{v}}\)
(iii)   $\vec{w} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$.
````{dropdown} Solution (click to show)
$$ \hat{\vec{w}} = \dfrac{\vec{w}}{|\vec{w}|} = \dfrac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} \\ 0 \\ \frac{\sqrt{2}}{2} \end{pmatrix} $$
Check magnitude of $\hat{\vec{w}}$
$$ \begin{align*}
|\hat{\vec{w}}| &= \sqrt{ \left( \frac{\sqrt{2}}{2} \right)^2 + 0^2 + \left( \frac{\sqrt{2}}{2} \right)^2 } \\
&= \sqrt{ \frac{2}{4} + 0 + \frac{2}{4} } = \sqrt{1} = 1 \qquad \checkmark
\end{align*} $$