leeern by Yuri

Notas de YuriRod

Página tipo blog en el que voy a publicar mis notas de aprendizaje, en especial de temas como matemáticas, física y quizá algo de programación

Redes neuronales
Redes neuronales
Redes neuronales
Redes neuronales

Vectores


θ=arg(v→)=0.14π rad\theta = arg(\overrightarrow{v}) = 0.14 \pi \ rad∣∣v→∣∣=2.23|| \overrightarrow{v}|| = 2.23
Algunas formas de representar al vector:

- Par ordenado:

v→=(2,1)\overrightarrow{v} = (2,1)

- Matriz:

v→=[21]\overrightarrow{v} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}

- Vectores canónicos:

v→=2i+1j\overrightarrow{v} = 2i + 1j

- Polar:

v→=∣v→∣θ=∣v→∣0.14π rad\overrightarrow{v}=|\overrightarrow{v}|_{\theta}=|\overrightarrow{v}|_{0.14 \pi \ rad}


Vectores unitarios

Vectores de módulo 1

v→∣∣v→∣∣\frac{\overrightarrow{v}}{||\overrightarrow{v}||}
Vectores canónicos
ı^=(1,0)\hat{\imath}=(1,0)ȷ^=(0,1)\hat{\jmath}=(0,1)
Ejemplo de vectores unitarios de v→,u→,w→\overrightarrow{v}, \overrightarrow{u}, \overrightarrow{w}

Operadores


Suma de vectores
v→+u→=w→ [−21]+[32]=[13]\textcolor{#3A92FF}{\overrightarrow{v}}+\textcolor{#52D1AB}{\overrightarrow{u}}=\textcolor{#A53DFC}{\overrightarrow{w}} \\ \ \\ \begin{bmatrix} \textcolor{#3A92FF}{-2} \\ \textcolor{#3A92FF}{1} \end{bmatrix}+ \begin{bmatrix} \textcolor{#52D1AB}{3} \\ \textcolor{#52D1AB}{2} \end{bmatrix} = \begin{bmatrix} \textcolor{#A53DFC}{1} \\ \textcolor{#A53DFC}{3} \end{bmatrix}

Producto con un escalar
2×v→ 2×[−21]=[−42]2 \times \textcolor{#52D1AB}{\overrightarrow{v}} \\ \ \\ 2 \times\begin{bmatrix} \textcolor{#52D1AB}{-2} \\ \textcolor{#52D1AB}{1} \end{bmatrix} = \begin{bmatrix} \textcolor{#3DBBE9}{-4} \\ \textcolor{#3DBBE9}{2} \end{bmatrix}
Escalar:2

Producto punto
v→⋅u→[32]⋅[−21]=(3×−2)+(2×1)=−4.00∥Proju→v→×∥v→∥∥=∥cos⁡(θ)×∥v→∥×u^⏞Proju→v→×∥u→∥∥=cos⁡(θ)×∥v→∥×∥u→∥∥Projv→u→×∥u→∥∥=∥cos⁡(θ)×∥u→∥×v^⏞Projv→u→×∥v→∥∥=cos⁡(θ)×∥u→∥×∥v→∥ \textcolor{#52D1AB}{\overrightarrow{v}} \cdot \textcolor{#3DBBE9}{\overrightarrow{u}} \\ \begin{bmatrix} \textcolor{#52D1AB}{3} \\ \textcolor{#52D1AB}{2} \end{bmatrix} \cdot \begin{bmatrix} \textcolor{#3DBBE9}{-2} \\ \textcolor{#3DBBE9}{1} \end{bmatrix} = (\textcolor{#52D1AB}{3} \times \textcolor{#3DBBE9}{-2}) + (\textcolor{#52D1AB}{2} \times \textcolor{#3DBBE9}{1}) = -4.00 \\ \lVert \textcolor{#EEA133}{\textit{Proj}_{\overrightarrow{u}}\overrightarrow{v}} \times \lVert \textcolor{#52D1AB}{\overrightarrow{v}}\rVert \rVert = \lVert \overset{ \textcolor{#EEA133}{\textit{Proj}_{\overrightarrow{u}}\overrightarrow{v}} }{ \overbrace{ \textcolor{#ff00b0}{\cos(\theta)} \times \lVert \textcolor{#52D1AB}{\overrightarrow{v}}\rVert \times \hat{u} } } \times \lVert \textcolor{#3DBBE9}{\overrightarrow{u}}\rVert \rVert = \textcolor{#ff00b0}{\cos(\theta)} \times \lVert \textcolor{#52D1AB}{\overrightarrow{v}}\rVert \times \lVert \textcolor{#3DBBE9}{\overrightarrow{u}}\rVert \\ \lVert \textcolor{#BCEE33}{\textit{Proj}_{\overrightarrow{v}}\overrightarrow{u}}\times \lVert \textcolor{#3DBBE9}{\overrightarrow{u}}\rVert \rVert = \lVert \overset{ \textcolor{#BCEE33}{\textit{Proj}_{\overrightarrow{v}}\overrightarrow{u}} }{ \overbrace{ \textcolor{#ff00b0}{\cos(\theta)} \times \lVert \textcolor{#3DBBE9}{\overrightarrow{u}}\rVert \times \hat{v} } } \times \lVert \textcolor{#52D1AB}{\overrightarrow{v}}\rVert \rVert = \textcolor{#ff00b0}{\cos(\theta)} \times \lVert \textcolor{#3DBBE9}{\overrightarrow{u}}\rVert \times \lVert \textcolor{#52D1AB}{\overrightarrow{v}}\rVert \\

Producto cruz
v→×u→=7u→×v→=−7v→⋅u→[320]×[−210]=(3×−2)+(2×1)=−4.00 \textcolor{#52D1AB}{\overrightarrow{v}}\times\textcolor{#3DBBE9}{\overrightarrow{u}}=7 \\ \textcolor{#3DBBE9}{\overrightarrow{u}}\times\textcolor{#52D1AB}{\overrightarrow{v}}=-7 \\ \textcolor{#52D1AB}{\overrightarrow{v}} \cdot \textcolor{#3DBBE9}{\overrightarrow{u}} \\ \begin{bmatrix} \textcolor{#52D1AB}{3} \\ \textcolor{#52D1AB}{2} \\ 0 \end{bmatrix} \times \begin{bmatrix} \textcolor{#3DBBE9}{-2} \\ \textcolor{#3DBBE9}{1} \\ 0 \end{bmatrix} = (\textcolor{#52D1AB}{3} \times \textcolor{#3DBBE9}{-2}) + (\textcolor{#52D1AB}{2} \times \textcolor{#3DBBE9}{1}) = -4.00

Producto Cruz

Proyección y componente ortogonal