Vectors

• 1D, 2D, 3D, 4D

Typical notation:

$$ \begin{align} y =& \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{m} \end{bmatrix} \ \ \ \ \bold{a} = \begin{bmatrix} a_x \\ a_y \\ a_z\end{bmatrix} \end{align} $$

Meaning :

💡 A vector has a meaning only if it is provided a context! (an origin)

• Position
• Direction
• Length : $|\bold{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$
• unit vector : $|\bold{a}| = 1$

Operations :

• Negative
• Dot Product
• Cross Product

In 2D the output of the cross product will be the area between two vectors and it will have no direction as there is no extra dimension. However in 3D it will have a direction that is perpendicular to both vectors, in the direction of the right hand rule.

Matrices