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Linear Algebra

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Coding Projects Header
Common

$$ a \cdot b = a_1b_1 + a_2b_2 + a_3b_3 + ... + a_nb_n \\\quad\bullet\quad a \cdot b = |a||b|cos\theta $$

$$ For\,a = \langle a_1, a_2, a_3\rangle\,and\,b = \langle b_1, b_2, b_3\rangle \rightarrow a \times b = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2, a_2b_1\rangle \\\quad\bullet\quad \text{Only for 3D vectors} $$

Distances
\( \text{Point-Point} \)\( |\overrightarrow{PQ}| \)\( \text{* In these cases, }|\overrightarrow{v}| \text{ implies } ||\overrightarrow{v}|| \)
\( \text{Point-Plane} \)\( \dfrac{|\overrightarrow{PQ}\,\cdot\,n|}{|n|} \)\( \text{P is the point, Q is some point on the plane, and n is the normal vector of that plane} \)
\( \text{Point-Line} \)\( \dfrac{|(\overrightarrow{PQ}) \times u|}{|u|} \)\( \text{P is the point and the line } \overrightarrow{r}(t) = Q + t\overrightarrow{u} \)
\( \text{Line-Line} \)\( \dfrac{|(\overrightarrow{PQ}) \cdot (u \times v)|}{|u \times v|} \)\( \overrightarrow{r}(t) = Q + t\overrightarrow{u} \quad\&\quad \overrightarrow{s}(t) = P + t\overrightarrow{v} \)
\( \text{Plane-Plane} \)\( \dfrac{|e - d|}{|n|} \)\( \text{If} \quad n \cdot x = d \quad\&\quad n \cdot x = e \quad \text{are parallel; otherwise it would be 0} \)