Vector Calculus (9) - Position Vector, Unit Vector

21. Position Vector

$ \vec{A}=(x_{1},x_{2},x_{3}) $

$ \vec{B}=(\bar{x_{1}},\bar{x_{2}},\bar{x_{3}}) $

 

$ |\vec{A}|=\sqrt{\vec{A}\cdot \vec{A}}=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}=\sqrt{\sum_{i}^{}x_{i}^{2}} $: Distance from the Origin to the Point $ (x_{1},x_{2},x_{3}) $

$ \sqrt{\sum_{i}^{}(x_{i}-\bar{x_{i}})^{2}}=\sqrt{(\vec{A}-\vec{B})\cdot(\vec{A}-\vec{B})}=|\vec{A}-\vec{B}| $: Distance from the point $ (x_{1},x_{2},x_{3}) $ to point $ (\bar{x_{1}},\bar{x_{2}},\bar{x_{3}}) $

 

We can define vector as the difference of the position vectors.

Orthogonal transformations are distance/angle-preserving transformations.

 

 

22. Unit Vector

 

$ \hat{e_{R}}=\frac{\vec{R}}{|\vec{R}|} $

Symbols of Unit Vectors: $ (\vec{i},\vec{j},\vec{k}), (\hat{e_{1}},\hat{e_{2}},\hat{e_{3}}), (\hat{e_{r}},\hat{e_{\Theta }},\hat{e_{\phi }}),(\vec{r},\vec{\Theta },\vec{\phi }) $ 등등

 

$ \vec{A}=(A_{1},A_{2},A_{3})=\hat{e_{1}}A_{1}+\hat{e_{2}}A_{2}+\hat{e_{3}}A_{2}=A_{1}\vec{i},A_{2}\vec{j},A_{3}\vec{k} $

$ A_{i}=\hat{e_{i}}\cdot \vec{A} $

 

if two vectors are orthogonal

$ \hat{e_{i}}\cdot \hat{e_{j}}=\delta _{ij} $

because $ i=j \Rightarrow 1, i\neq j\Rightarrow 0 $