Chapter 4 Vectors Notation
4.1 Rectangular Vectors
A vector \(v\) is composed of n-components of n-dimensionality. For example, a 3-dimensional vector will have 3 components, commonly represented as \(x\), \(y\), and \(z\), or \(i\), \(j\), and \(k\).
\[ v \triangleq [v_1, v_2, v_3, ... v_n]\]
A rectangular vector can be specified as the sum of the scalar multiples of the components of the vector with the members of the standard basis or reference axes.
The basis is represented with unit vectors \[\begin{equation} \begin{split} \hat{i} & =(1,0,0)\\ \hat{j} & =(0,1,0)\\ \hat{k} & =(0,0,1) \end{split} \tag{4.1} \end{equation}\]
In rectangular vector notation, we could write this as
\[\begin{equation} v = v_x\hat{i}+v_y\hat{j}+v_z\hat{k} \tag{4.2} \end{equation}\]
4.2 Directional Vectors
An alternative representation of vectors is to represent a vector as a direction and magnitude
\[v = r\measuredangle \psi\]
where \(r\) is the magnitude and \(\psi\) represents the direction of the vector.
This representation will not be used in this class, but is good to remember.
4.3 Norms
4.3.2 Euclidean Distance Norm - L2
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\[\Theta = \begin{pmatrix}\alpha & \beta\\ \gamma & \delta \end{pmatrix}\]
$$\Theta = \begin{pmatrix}\alpha & \beta\\
\gamma & \delta
\end{pmatrix}$$\[\begin{equation*} \frac{d}{dx}\left( \int_{a}^{x} f(u)\,du\right)=f(x) \end{equation*}\]