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Vector Geometry.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} {\large Vector Geometry} \begin{align*} \text{Vector:}\quad&\text{A quantity which has \it magnitude \rm and \it direction}\\ &\text{On the Argand Diagram, a vector is an arrow-head \--- length represents magnitude, head indicates direction.}\\ &\text{If a complex number $a+ib$ is represented as point $P(a,b)$, then the vector can be drawn from the origin as }\overrightarrow{OP}\:.\\ &\text{Vectors have no ``position'', and can be drawn from anywhere on the Argand Diagram, e.g. }\overrightarrow{AB}\:.\\ &\text{Two vectors are equal if they are of the same \it magnitude \rm and pointing to the same \it direction\rm,}\\ &\qquad\text{so $\overrightarrow{OP}=\overrightarrow{AB}$ if they are of the same length (magnitude) and gradient (direction).}\\ \\ \text{In this context:}\quad&z=a+ib=r\:(\cos\theta+i\sin\theta)\neq 0\text{ , and is represented by }\overrightarrow{OP}\text{, of which the ``off-the-origin'' equivalent is }\overrightarrow{AB}\:.\\ &z_m=a_m+ib_m=r_m\:(\cos\theta_m+i\sin\theta_m)\neq 0\text{ , and is represented by }\overrightarrow{OP_m}\:,\\ &\qquad\text{of which the ``off-the-origin'' equivalent is represented by }\overrightarrow{A_mB_m}\:,\quad\text{where }m=1,2,3,\ldots\\ \\ \text{Modulas and }&\text{Argument:}\\ &\text{The modulas }|z|=r=\sqrt{a^2+b^2}\text{ is the length of $OP$ .}\\ &\text{The (principal) argument }\arg(z)=\theta=\tan^{-1}\tfrac{b}{a}\text{ (or $\pm\tfrac{\pi}{2}$ when $a=0$) is the smaller angle of $\angle POX$ ,}\\ &\qquad\text{where $OX$ is the positive x-axis and the sign of $\angle POX$ is the same as the sign of $b$.}\\ \\ &\arg\left(\tfrac{z_1}{z_2}\right)=\theta\quad\text{if $OP_2$ can rotate anticlockwise less than a semi-circle to reach $OP_1$;}\\ &\arg\left(\tfrac{z_1}{z_2}\right)=-\theta\quad\text{if $OP_2$ can rotate clockwise less than a semi-circle to reach $OP_1$;}\\ &\arg\left(\tfrac{z_1}{z_2}\right)=\pi\quad\text{if $O$ is on line segment $P_1P_2$ internally. ($\arg(\tfrac{z_1}{z_2})=0$ if $O$ is on $P_1P_2$ externally.)}\\ \\ \text{Transformation:}\quad &\text{$z_1+z_2$ is $\overrightarrow{OP_2'}$, where $\overrightarrow{P_1P_2'}=\overrightarrow{OP_2}=z_2$ .}\\ &\text{$-z_2$ is $\overrightarrow{OP_3}$, where $O$ is the midpoint of $P_2P_3$ .}\\ &\text{$z_1-z_2$ is $\overrightarrow{OP_3'}$, where $\overrightarrow{P_1P_3'}=\overrightarrow{OP_3}=-z_2$ .}\\ &\text{$z_1z_2$ is a vector obtained by \it enlarging \rm $\overrightarrow{OP_1}$ by $r_2$ and rotating the vector \it anticlockwise \rm by $\theta_2$ .}\\ &\text{$\frac{z_1}{z_2}$ is a vector obtained by \it shrinking \rm $\overrightarrow{OP_1}$ by $r_2$ and rotating the vector \it clockwise \rm by $\theta_2$ .}\\ \\ \text{Properties:}\quad&\text{Geometry Properties derivable from Complex Number Relations}\\ &\boxed{\text{A \it positional \rm off-the-origin ``arrow'' $\overrightarrow{P_1P_2}$ must be represented by $z_2-z_1$, $P_2$ head, $P_1$ tail.}}\\ &\arg\left(\tfrac{z_1}{z_2}\right)=0\text{ or }\pi\:,\quad\text{i.e. }z_1=kz_2,\text{ where $k$ is real and }k\neq 0\quad\Rightarrow\quad A_1B_1\parallel A_2B_2\\ &\arg\left(\tfrac{z_1}{z_2}\right)=\pm\frac{\pi}{2}\:,\quad\text{i.e. }\frac{z_1}{z_2}=k\:i,\text{ where $k$ is real and }k\neq 0\quad\Rightarrow\quad A_1B_1\perp A_2B_2\\ &\text{If $A_1$ and $A_2$ coincide at $A$, $\angle B_1AB_2=\left|arg\left(\tfrac{z_1}{z_2}\right)\right|=\left|arg\left(\tfrac{z_2}{z_1}\right)\right|$ .}\\ &\text{The midpoint of $P_1P_2$ is }\frac{z_1+z_2}{2}\:.\\ \end{align*} \end{document}