616033b32f0e37bcaa14653f7bec259f64ad5a24
1\documentclass[a4paper]{article}
2\usepackage[a4paper,margin=2cm]{geometry}
3\usepackage{multicol}
4\usepackage{multirow}
5\usepackage{amsmath}
6\usepackage{amssymb}
7\usepackage{harpoon}
8\usepackage{tabularx}
9\usepackage[dvipsnames, table]{xcolor}
10\usepackage{graphicx}
11\usepackage{wrapfig}
12\usepackage{tikz}
13\usepackage{tikz-3dplot}
14\usetikzlibrary{calc}
15\usetikzlibrary{angles}
16\usepgflibrary{arrows.meta}
17\usepackage{fancyhdr}
18\pagestyle{fancy}
19\fancyhead[LO,LE]{Year 12 Specialist}
20\fancyhead[CO,CE]{Andrew Lorimer}
21
22\usepackage{mathtools}
23\usepackage{xcolor} % used only to show the phantomed stuff
24\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
25\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
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28\newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
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30\definecolor{cas}{HTML}{e6f0fe}
31\linespread{1.5}
32\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
33
34\begin{document}
35
36\begin{multicols}{2}
37
38 \section{Complex numbers}
39
40 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
41
42 \begin{align*}
43 \text{Cartesian form: } & a+bi\\
44 \text{Polar form: } & r\operatorname{cis}\theta
45 \end{align*}
46
47 \subsection*{Operations}
48
49\definecolor{shade1}{HTML}{ffffff}
50\definecolor{shade2}{HTML}{e6f2ff}
51 \definecolor{shade3}{HTML}{cce2ff}
52 \begin{tabularx}{\columnwidth}{r|X|X}
53 & \textbf{Cartesian} & \textbf{Polar} \\
54 \hline
55 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
56 \hline
57 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
58 \cline{1-1}\cline{3-3}
59 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
60 \hline
61 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
62 \hline
63 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
64 \end{tabularx}
65
66 \subsubsection*{Scalar multiplication in polar form}
67
68 For \(k \in \mathbb{R}^+\):
69 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
70
71 \noindent For \(k \in \mathbb{R}^-\):
72 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
73
74 \subsection*{Conjugate}
75
76 \begin{align*}
77 \overline{z} &= a \mp bi\\
78 &= r \operatorname{cis}(-\theta)
79 \end{align*}
80
81 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
82
83 \subsubsection*{Properties}
84
85 \begin{align*}
86 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
87 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
88 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
89 z\overline{z} &= (a+bi)(a-bi)\\
90 &= a^2 + b^2\\
91 &= |z|^2
92 \end{align*}
93
94 \subsection*{Modulus}
95
96 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
97
98 \subsubsection*{Properties}
99
100 \begin{align*}
101 |z_1z_2|&=|z_1||z_2|\\
102 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
103 |z_1+z_2|&\le|z_1|+|z_2|
104 \end{align*}
105
106 \subsection*{Multiplicative inverse}
107
108 \begin{align*}
109 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
110 &=\frac{\overline{z}}{|z|^2}a\\
111 &=r \operatorname{cis}(-\theta)
112 \end{align*}
113
114 \subsection*{Dividing over \(\mathbb{C}\)}
115
116 \begin{align*}
117 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
118 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
119 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
120 & \qquad \text{(rationalise denominator)}
121 \end{align*}
122
123 \subsection*{Polar form}
124
125 \begin{align*}
126 z&=r\operatorname{cis}\theta\\
127 &=r(\cos \theta + i \sin \theta)
128 \end{align*}
129
130 \begin{itemize}
131 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
132 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
133 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
134 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
135 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
136 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
137 \end{itemize}
138
139 \subsection*{de Moivres' theorem}
140
141 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
142
143 \subsection*{Complex polynomials}
144
145 Include \(\pm\) for all solutions, incl. imaginary
146
147 \begin{tabularx}{\columnwidth}{ R{0.55} X }
148 \hline
149 Sum of squares & \(\begin{aligned}
150 z^2 + a^2 &= z^2-(ai)^2\\
151 &= (z+ai)(z-ai) \end{aligned}\) \\
152 \hline
153 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
154 \hline
155 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
156 \hline
157 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
158 \hline
159 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
160 \hline
161 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)
162 \end{tabularx}
163
164 \subsection*{Roots}
165
166 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
167
168 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
169
170 \begin{itemize}
171
172 \item{Same modulus for all solutions}
173 \item{Arguments are separated by \(\frac{2\pi}{n}\)}
174 \item{Solutions of \(z^n=a\) where \(a \in \mathbb{C}\) lie on the circle \(x^2+y^2=\left(|a|^{\frac{1}{n}}\right)^2\) \quad (intervals of \(\frac{2\pi}{n}\))}
175 \end{itemize}
176
177 \noindent For \(0=az^2+bz+c\), use quadratic formula:
178
179 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
180
181 \subsection*{Fundamental theorem of algebra}
182
183 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
184
185 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
186 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
187
188 \subsection*{Argand planes}
189
190 \begin{center}\begin{tikzpicture}[scale=2]
191 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
192 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
193 \coordinate (P) at (1,1);
194 \coordinate (a) at (1,0);
195 \coordinate (b) at (0,1);
196 \coordinate (O) at (0,0);
197 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
198 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
199 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
200 \begin{scope}
201 \path[clip] (O) -- (P) -- (a);
202 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
203 \node at ($(O)+(20:3mm)$) {$\theta$};
204 \end{scope}
205 \filldraw (P) circle (0.5pt);
206 \end{tikzpicture}\end{center}
207
208 \begin{itemize}
209 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
210 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
211 \end{itemize}
212
213 \subsection*{Sketching complex graphs}
214
215 \subsubsection*{Linear}
216
217 \begin{itemize}
218 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
219 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
220 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)}
221 \end{itemize}
222
223 \subsubsection*{Circles}
224
225 \begin{itemize}
226 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
227 \item \(|z-(a+bi)|=c\)
228 \end{itemize}
229
230 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
231
232 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
233 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
234 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
235 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
236 \draw [->, thick, blue] (0,0) -- (1,1);
237 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
238 \begin{scope}
239 \path[clip] (0,0) -- (1,1) -- (1,0);
240 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
241 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
242 \end{scope}
243 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
244 \node [blue, mydot] {};
245 \end{tikzpicture}\end{center}
246
247 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
248
249 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
250 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
251 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
252 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
253 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
254 \begin{scope}
255 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
256 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
257 \end{scope}
258 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
259 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
260 \node [brown, mydot] at (-0.25,0) {};
261 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
262 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
263 \node [below, font=\footnotesize] at (1,0) {\(1\)};
264 \end{tikzpicture}\end{center}
265
266 \section{Vectors}
267\begin{center}\begin{tikzpicture}
268 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
269 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
270 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
271 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
272 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
273 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
274 \end{scope}
275 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
276
277\end{tikzpicture}\end{center}
278
279\subsection*{Column notation}
280
281\[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
282\(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
283
284\subsection*{Scalar multiplication}
285
286\[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
287
288\noindent For \(k \in \mathbb{R}^-\), direction is reversed
289
290\subsection*{Vector addition}
291\begin{center}\begin{tikzpicture}[scale=1]
292 \coordinate (A) at (0,0);
293 \coordinate (B) at (2,2);
294 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
295 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
296 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
297\end{tikzpicture}\end{center}
298
299\[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
300
301\begin{itemize}
302 \item Draw each vector head to tail then join lines
303 \item Addition is commutative (parallelogram)
304 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v})\)
305\end{itemize}
306
307\subsection*{Magnitude}
308
309\[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
310
311\subsection*{Parallel vectors}
312
313\[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
314
315For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
316\[\boldsymbol{a \cdot b}=\begin{cases}
317|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
318-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
319\end{cases}\]
320%\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
321%\includegraphics[width=1]{graphics/vector-subtraction.jpg}
322
323\subsection*{Perpendicular vectors}
324
325\[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
326
327\subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
328\[\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
329
330 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
331
332
333\begin{center}\begin{tikzpicture}[scale=2]
334 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
335 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
336 \begin{scope}
337 \path[clip] (1,0.5) -- (1,0) -- (0,0);
338 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
339 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
340 \end{scope}
341\end{tikzpicture}\end{center}
342\begin{align*}\boldsymbol{a} \cdot \boldsymbol{b} &= a_1 b_1 + a_2 b_2 \\ &= |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta \\ &\quad (\> 0 \le \theta \le \pi) \text{ - from cosine rule}\end{align*}
343\noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
344
345\subsubsection*{Properties}
346
347\begin{enumerate}
348\item
349 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
350\item
351 \(\boldsymbol{a \cdot 0}=0\)
352\item
353 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
354\item
355 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
356\item
357 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
358\item
359 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
360\end{enumerate}
361
362\subsection*{Angle between vectors}
363
364\[\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}\]
365
366\noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
367
368(Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
369
370\subsection*{Angle between vector and axis}
371
372\noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
373which makes angles \(\alpha, \beta, \gamma\) with positive side of
374\(x, y, z\) axes:
375\[\cos \alpha = {a_1 \over |\boldsymbol{a}|}, \quad \cos \beta = {a_2 \over |\boldsymbol{a}|}, \quad \cos \gamma = {a_3 \over |\boldsymbol{a}|}\]
376
377\noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
378between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
379\(x\)-axis
380
381\subsection*{Projections \& resolutes}
382
383\begin{tikzpicture}[scale=3]
384 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
385 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
386 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
387 \begin{scope}
388 \path[clip] (1,0.5) -- (1,0) -- (0,0);
389 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
390 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
391 \end{scope}
392 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
393 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
394 \end{scope}
395\draw (0,0) coordinate (O)
396 (1,0) coordinate (A)
397 (1,0.5) coordinate (B)
398 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
399\end{tikzpicture}
400
401\subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
402\begin{align*}
403 \boldsymbol{u}&={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}\\
404 &=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)\\
405 &=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
406\end{align*}
407
408\subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
409\[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
410
411\subsubsection*{\(|\boldsymbol{u}|\) (scalar resolute)}
412\begin{align*}
413 r_s &= |\boldsymbol{u}|\\
414 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
415 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}
416\end{align*}
417
418\subsubsection*{Rectangular (\(\parallel,\perp\)) components}
419
420\[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
421
422
423\subsection*{Vector proofs}
424
425\textbf{Concurrent:} intersection of \(\ge\) 3 lines
426
427\begin{tikzpicture}
428 \draw [blue] (0,0) -- (1,1);
429 \draw [red] (1,0) -- (0,1);
430 \draw [brown] (0.4,0) -- (0.6,1);
431 \filldraw (0.5,0.5) circle (2pt);
432\end{tikzpicture}
433
434\subsubsection*{Collinear points}
435
436\(\ge\) 3 points lie on the same line
437
438\begin{tikzpicture}
439 \draw [purple] (0,0) -- (4,1);
440 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
441 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
442 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
443 \coordinate (O) at (2.8,-0.2);
444 \node at (O) [below] {\(O\)};
445 \begin{scope}[->, orange, thick]
446 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
447 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
448 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
449 \end{scope}
450\end{tikzpicture}
451
452\begin{align*}
453 \text{e.g. Prove that}\\
454 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
455 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
456 &= \overrightharp{OA} + m\overrightharp{AB}\\
457 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
458 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
459 &=(1-m)\boldsymbol{a}+m{b}
460\end{align*}
461
462\begin{align*}
463 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
464 \text{where } \lambda + \mu &= 1\\
465 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
466\end{align*}
467
468
469 \subsubsection*{Useful vector properties}
470
471\begin{itemize}
472\item
473 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel, then
474 \(\boldsymbol{b}=k\boldsymbol{a}\) for some
475 \(k \in \mathbb{R} \setminus \{0\}\)
476\item
477 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
478 least one point in common, then they lie on the same straight line
479\item
480 Two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are
481 perpendicular if \(\boldsymbol{a} \cdot \boldsymbol{b}=0\)
482\item
483 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
484\end{itemize}
485
486\subsection*{Linear dependence}
487
488Vectors \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly
489dependent if they are non-parallel and:
490
491\[k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0\]
492\[\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}\]
493
494\(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
495independent if no vector in the set is expressible as a linear
496combination of other vectors in set, or if they are parallel.
497
498Vector \(\boldsymbol{w}\) is a linear combination of vectors
499\(\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}\)
500
501\subsection*{Three-dimensional vectors}
502
503Right-hand rule for axes: \(z\) is up or out of page.
504
505\tdplotsetmaincoords{60}{120}
506\begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
507vector/.style={-stealth,red,very thick},
508vector guide/.style={dashed,gray,thick}]
509
510%standard tikz coordinate definition using x, y, z coords
511\coordinate (O) at (0,0,0);
512
513%tikz-3dplot coordinate definition using x, y, z coords
514
515\pgfmathsetmacro{\ax}{1}
516\pgfmathsetmacro{\ay}{1}
517\pgfmathsetmacro{\az}{1}
518
519\coordinate (P) at (\ax,\ay,\az);
520
521%draw axes
522\draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
523\draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
524\draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
525
526%draw a vector from O to P
527\draw[vector] (O) -- (P);
528
529%draw guide lines to components
530\draw[vector guide] (O) -- (\ax,\ay,0);
531\draw[vector guide] (\ax,\ay,0) -- (P);
532\draw[vector guide] (P) -- (0,0,\az);
533\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
534\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
535\draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
536\node[tdplot_main_coords,above right]
537at (\ax,\ay,\az){(\ax, \ay, \az)};
538\end{tikzpicture}\end{center}
539
540\subsection*{Parametric vectors}
541
542Parametric equation of line through point \((x_0, y_0, z_0)\) and
543parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
544
545\begin{equation}\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\end{equation}
546
547
548 \end{multicols}
549\end{document}