User:Laz-Roc

= Vectors =

Column & Row Vectors
$$\mathbf{a} = [ a\ b\ c ]$$ $$\mathbf{a} = \begin{bmatrix} a\\ b\\ c\\ \end{bmatrix} $$

Magnitude
$$\left\|\mathbf{v}\right\|=\sqrt{v_x^2+v_y^2+v_z^2}$$

Vector-Scalar Multiplication
$$\mathbf{k} = \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} kx\\ ky\\ kz\\ \end{bmatrix} $$

$$\mathbf{\frac{v}{k}}\ = \left ( \frac{1}{k} \right ) v = \begin{bmatrix} v_x/k\\ v_y/k\\ v_z/k\\ \end{bmatrix}$$

Normalizing Vectors
$$ \mathbf{v_{norm}} = \frac{v} { \left\| v \right\| }, v \ne 0 $$

Vector Addition & Subtraction
$$\begin{bmatrix} x_1\\ y_1\\ z_1\\ \end{bmatrix} + \begin{bmatrix} x_2\\ y_2\\ z_2\\ \end{bmatrix} = \begin{bmatrix} x_1 + x_2\\ y_1 + y_2\\ z_1 + z_2\\ \end{bmatrix} $$

$$\begin{bmatrix} x_1\\ y_1\\ z_1\\ \end{bmatrix} - \begin{bmatrix} x_2\\ y_2\\ z_2\\ \end{bmatrix} = \begin{bmatrix} x_1\\ y_1\\ z_1\\ \end{bmatrix} + \left ( \begin{bmatrix} x_2\\ y_2\\ z_2\\ \end{bmatrix} \right ) = \begin{bmatrix} x_1 - x_2\\ y_1 - y_2\\ z_1 - z_2\\ \end{bmatrix} $$

Distance
$$ \mathbf{d} = b-a = \begin{bmatrix} b_x - a_x\\ b_y - a_y\\ b_z - a_z\\ \end{bmatrix} $$

$$ distance \mathbf{(a,b)} = \left\|\mathbf{d}\right\| = \sqrt{d_x^2 + d_y^2 + d_z^2} $$

Dot Product
$$ \mathbf{a \cdot b} = a_x b_x + a_y b_y + a_z b_z $$

$$ \mathbf{a \cdot b} = \left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\| \cos \theta $$

$$ \mathbf{\theta} = \arccos \left ( \frac{a\cdot b}{\left\|a\right\| \left\|b\right\|} \right ) $$

$$ v_1 = n \frac {\left\| v_1 \right\|} {\left\| n \right\|} $$

$$ \mathbf{\cos\theta} = \frac{\left\|v_1\right\|}{\left\|v_1\right\|}, \mathbf{\cos\theta\left\|v\right\|} = \left\|v\right\| $$

$$ \mathbf{v_1} = n \frac {\left\|v\right\|\cos\theta} {\left\|n\right\|} = n \frac {\left\|v\right\|\left\|n\right\|\cos\theta}{\left\|n\right\|^2} = n \frac {v \cdot n} {\left\|n\right\|^2} $$

$$ \mathbf{v_2 + v_1} = \left\|v\right\| $$

$$ \mathbf{v_2} = \left\|v\right\| - v_1 = \left\|v\right\| - n \frac {v \cdot n} {\left\|n\right\|^2} $$

Cross Product
$$ \begin{bmatrix} x_1\\ y_2\\ z_3\\ \end{bmatrix} \times \begin{bmatrix} x_2\\ y_2\\ z_2\\ \end{bmatrix} = \begin{bmatrix} y_1 z_2 - z_1 y_2\\ z_1 x_2 - x_1 z_2\\ x_1 y_2 - y_1 x_2\\ \end{bmatrix} $$

$$ \mathbf{\left\|a \times b \right\|} = \left\|a\right\| \left\|b\right\| \sin \theta $$

$$ \mathbf{A} = bh = b \left ( a \sin \theta \right ) = \left\|a\right\| \left\|b\right\| \sin \theta = \left\|a \times b \right\| $$