$\overrightarrow{\text{}v\text{}}\xb7\overrightarrow{\text{}w\text{}}=\overrightarrow{\text{}v\text{}}\overrightarrow{\text{}w\text{}}\mathrm{cos}\theta$  $(1)$ 
$\overrightarrow{\text{}v\text{}}\xb7\overrightarrow{\text{}v\text{}}=\overrightarrow{\text{}v\text{}}{}^{2}$  $(2)$ 
$\hat{\text{}\u0131\text{}}\xb7\hat{\text{}\u0131\text{}}=1$  $(3)$ 
$\overrightarrow{\text{}v\text{}}\perp \overrightarrow{\text{}w\text{}}\iff \overrightarrow{\text{}v\text{}}\xb7\overrightarrow{\text{}w\text{}}=0$  $(4)$ 
$\hat{\text{}\u0131\text{}}\xb7\hat{\text{}\u0237\text{}}=0$  $(5)$ 
$\begin{array}{ccc}\multicolumn{1}{c}{{v}_{x}}& =\hfill & \overrightarrow{\text{}v\text{}}\xb7\hat{\text{}\u0131\text{}}\hfill \\ \multicolumn{1}{c}{{v}_{y}}& =\hfill & \overrightarrow{\text{}v\text{}}\xb7\hat{\text{}\u0237\text{}}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(6)\end{array}$ 
$\begin{array}{ccc}\multicolumn{1}{c}{\overrightarrow{\text{}v\text{}}\xb7\overrightarrow{\text{}w\text{}}}& =\hfill & ({v}_{x}\hspace{0.5em}\hat{\text{}\u0131\text{}}+{v}_{y}\hspace{0.5em}\hat{\text{}\u0237\text{}})\xb7({w}_{x}\hspace{0.5em}\hat{\text{}\u0131\text{}}+{w}_{y}\hspace{0.5em}\hat{\text{}\u0237\text{}})\hfill \\ \multicolumn{1}{c}{}& =\hfill & {v}_{x}{w}_{x}\hspace{0.5em}\hat{\text{}\u0131\text{}}\xb7\hat{\text{}\u0131\text{}}+{v}_{y}{w}_{y}\hspace{0.5em}\hat{\text{}\u0237\text{}}\xb7\hat{\text{}\u0237\text{}}+{v}_{x}{w}_{y}\hspace{0.5em}\hat{\text{}\u0131\text{}}\xb7\hat{\text{}\u0237\text{}}+{v}_{y}{w}_{x}\hspace{0.5em}\hat{\text{}\u0237\text{}}\xb7\hat{\text{}\u0131\text{}}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(7)\\ \multicolumn{1}{c}{}& =\hfill & {v}_{x}{w}_{x}+{v}_{y}{w}_{y}\hfill \end{array}$ 
$\overrightarrow{\text{}v\text{}}\xb7\overrightarrow{\text{}v\text{}}=\overrightarrow{\text{}v\text{}}{}^{2}={v}_{x}^{2}+{v}_{y}^{2}$  $(8)$ 
$\begin{array}{ccc}\multicolumn{1}{c}{\overrightarrow{\text{}C\text{}}\xb7\overrightarrow{\text{}C\text{}}}& =\hfill & (\overrightarrow{\text{}A\text{}}+\overrightarrow{\text{}B\text{}})\xb7(\overrightarrow{\text{}A\text{}}+\overrightarrow{\text{}B\text{}})\hfill \\ \multicolumn{1}{c}{}& =\hfill & \overrightarrow{\text{}A\text{}}\xb7\overrightarrow{\text{}A\text{}}+\overrightarrow{\text{}B\text{}}\xb7\overrightarrow{\text{}B\text{}}2\hspace{0.5em}\overrightarrow{\text{}A\text{}}\xb7\overrightarrow{\text{}B\text{}}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(9)\end{array}$ 
$\overrightarrow{\text{}C\text{}}{}^{2}=\overrightarrow{\text{}A\text{}}{}^{2}+\overrightarrow{\text{}B\text{}}{}^{2}2\overrightarrow{\text{}A\text{}}\overrightarrow{\text{}B\text{}}\mathrm{cos}\theta$  $(10)$ 


$(\overrightarrow{\text{}v\text{}}+\overrightarrow{\text{}u\text{}})\xb7\overrightarrow{\text{}w\text{}}=\overrightarrow{\text{}v\text{}}\xb7\overrightarrow{\text{}w\text{}}+\overrightarrow{\text{}u\text{}}\xb7\overrightarrow{\text{}w\text{}}$  $(11)$ 




$\overrightarrow{\text{}v\text{}}\times \overrightarrow{\text{}w\text{}}=\overrightarrow{\text{}v\text{}}\overrightarrow{\text{}w\text{}}\mathrm{sin}\theta$  $(12)$ 
$\overrightarrow{\text{}v\text{}}\parallel \overrightarrow{\text{}w\text{}}\iff \overrightarrow{\text{}v\text{}}\times \overrightarrow{\text{}w\text{}}=\overrightarrow{0}$  $(13)$ 
$\overrightarrow{\text{}v\text{}}\times \overrightarrow{\text{}w\text{}}=\overrightarrow{\text{}w\text{}}\times \overrightarrow{\text{}v\text{}}$  $(14)$ 
$\overrightarrow{\text{}v\text{}}\times \overrightarrow{\text{}v\text{}}=\overrightarrow{0}$  $(15)$ 
$\begin{array}{ccc}\multicolumn{1}{c}{\hat{\text{}\u0131\text{}}\times \hat{\text{}\u0237\text{}}}& =\hfill & \hat{\text{}k\text{}}\hfill \\ \multicolumn{1}{c}{\hat{\text{}\u0237\text{}}\times \hat{\text{}k\text{}}}& =\hfill & \hat{\text{}\u0131\text{}}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(16)\\ \multicolumn{1}{c}{\hat{\text{}k\text{}}\times \hat{\text{}\u0131\text{}}}& =\hfill & \hat{\text{}\u0237\text{}}\hfill \end{array}$ 
$\begin{array}{ccc}\multicolumn{1}{c}{\overrightarrow{\text{}v\text{}}\times \overrightarrow{\text{}w\text{}}}& =\hfill & ({v}_{x}\hspace{0.5em}\hat{\text{}\u0131\text{}}+{v}_{y}\hspace{0.5em}\hat{\text{}\u0237\text{}}+{v}_{z}\hspace{0.5em}\hat{\text{}k\text{}})\times ({w}_{x}\hspace{0.5em}\hat{\text{}\u0131\text{}}+{w}_{y}\hspace{0.5em}\hat{\text{}\u0237\text{}}+{w}_{z}\hspace{0.5em}\hat{\text{}k\text{}})\hfill \\ \multicolumn{1}{c}{}& =\hfill & {v}_{x}{w}_{x}\hspace{0.5em}\hat{\text{}\u0131\text{}}\times \hat{\text{}\u0131\text{}}+{v}_{x}{w}_{y}\hspace{0.5em}\hat{\text{}\u0131\text{}}\times \hat{\text{}\u0237\text{}}+...\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(17)\\ \multicolumn{1}{c}{}& =\hfill & ({v}_{y}{w}_{z}{v}_{z}{w}_{y})\hspace{0.5em}\hat{\text{}\u0131\text{}}+({v}_{z}{w}_{x}{v}_{x}{w}_{z})\hspace{0.5em}\hat{\text{}\u0237\text{}}+({v}_{x}{w}_{y}{v}_{y}{w}_{x})\hspace{0.5em}\hat{\text{}k\text{}}\hfill \end{array}$ 
$\overrightarrow{\text{}v\text{}}\times \overrightarrow{\text{}w\text{}}=\left\begin{array}{ccc}\hfill \hat{\text{}\u0131\text{}}\hfill & \hfill \hat{\text{}\u0237\text{}}\hfill & \hfill \hat{\text{}k\text{}}\hfill \\ \multicolumn{1}{c}{{v}_{x}}& \hfill {v}_{y}\hfill & \hfill {v}_{z}\hfill \\ \multicolumn{1}{c}{{w}_{x}}& \hfill {w}_{y}\hfill & \hfill {w}_{z}\hfill \end{array}\right$  $(18)$ 


$\overrightarrow{\text{}w\text{}}\times (\overrightarrow{\text{}v\text{}}+\overrightarrow{\text{}u\text{}})=\overrightarrow{\text{}w\text{}}\times \overrightarrow{\text{}v\text{}}+\overrightarrow{\text{}w\text{}}\times \overrightarrow{\text{}u\text{}}$  $(19)$ 
$\text{}$  $(20)$ 
$\begin{array}{ccc}\multicolumn{1}{c}{\mathrm{cos}(\beta \alpha )}& =\hfill & \overrightarrow{\text{}u\text{}}\xb7\overrightarrow{\text{}v\text{}}=(\mathrm{cos}\alpha \hspace{0.5em}\hat{\text{}\u0131\text{}}+\mathrm{sin}\alpha \hspace{0.5em}\hat{\text{}\u0237\text{}})\xb7(\mathrm{cos}\beta \hspace{0.5em}\hat{\text{}\u0131\text{}}+\mathrm{sin}\beta \hspace{0.5em}\hat{\text{}\u0237\text{}})\hfill \\ \multicolumn{1}{c}{}& =\hfill & \mathrm{cos}\alpha \mathrm{cos}\beta +\mathrm{sin}\alpha \mathrm{sin}\beta \hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(21)\end{array}$ 