+ \begin{enumerate}
+ \item
+ \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
+ \item
+ \(\boldsymbol{a \cdot 0}=0\)
+ \item
+ \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
+ \item
+ \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
+ \item
+ \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
+ \item
+ \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
+ \end{enumerate}
+
+ \subsection*{Angle between vectors}
+
+ \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
+
+ \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
+
+ (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
+
+ \subsection*{Angle between vector and axis}
+
+ \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
+ which makes angles \(\alpha, \beta, \gamma\) with positive side of
+ \(x, y, z\) axes:
+ \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
+
+ \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
+ between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
+ \(x\)-axis
+
+ \subsection*{Projections \& resolutes}
+
+ \begin{tikzpicture}[scale=3]
+ \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
+ \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
+ \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
+ \begin{scope}
+ \path[clip] (1,0.5) -- (1,0) -- (0,0);
+ \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
+ \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
+ \end{scope}
+ \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
+ \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
+ \end{scope}
+ \draw (0,0) coordinate (O)
+ (1,0) coordinate (A)
+ (1,0.5) coordinate (B)
+ pic [draw,red,angle radius=2mm] {right angle = O--A--B};
+ \end{tikzpicture}
+
+ \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
+
+ \begin{align*}
+ \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
+ & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
+ & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
+ \end{align*}
+
+ \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
+ \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
+
+ \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
+ \begin{align*}
+ s &= |\boldsymbol{u}|\\
+ &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
+ &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
+ &= |\boldsymbol{a}| \cos \theta
+ \end{align*}
+
+ \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
+
+ \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
+
+
+ \subsection*{Vector proofs}
+
+ \textbf{Concurrent:} intersection of \(\ge\) 3 lines
+
+ \begin{tikzpicture}
+ \draw [blue] (0,0) -- (1,1);
+ \draw [red] (1,0) -- (0,1);
+ \draw [brown] (0.4,0) -- (0.6,1);
+ \filldraw (0.5,0.5) circle (2pt);
+ \end{tikzpicture}
+
+ \subsubsection*{Collinear points}
+
+ \(\ge\) 3 points lie on the same line
+
+ \begin{tikzpicture}
+ \draw [purple] (0,0) -- (4,1);
+ \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
+ \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
+ \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
+ \coordinate (O) at (2.8,-0.2);
+ \node at (O) [below] {\(O\)};
+ \begin{scope}[->, orange, thick]
+ \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
+ \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
+ \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
+ \end{scope}
+ \end{tikzpicture}
+
+ \begin{align*}
+ \text{e.g. Prove that}\\
+ \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
+ \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
+ &= \overrightharp{OA} + m\overrightharp{AB}\\
+ &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
+ &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
+ &=(1-m)\boldsymbol{a}+m{b}
+ \end{align*}
+ \begin{align*}
+ \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
+ \text{where } \lambda + \mu &= 1\\
+ \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
+ \end{align*}
+
+
+ \subsubsection*{Parallelograms}
+
+ \begin{center}\begin{tikzpicture}
+ \coordinate (O) at (0,0) node [below left] {\(O\)};
+ \coordinate (A) at (4,0);
+ \coordinate (B) at (6,2);
+ \coordinate (C) at (2,2);
+ \coordinate (D) at (6,0);
+
+ \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
+ \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
+ \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
+ \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
+
+ \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
+ \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
+ \begin{scope}
+ \path[clip] (C) -- (A) -- (O);
+ \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
+ \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
+ \end{scope}
+ \draw [gray, thick, dotted] (B) -- (D) node [pos=0.5, right, black, font=\footnotesize] {\(|\boldsymbol{c}|\sin\theta\)} (A) -- (D) node [pos=0.5, below, black, font=\footnotesize] {\(|\boldsymbol{c}|\cos\theta\)};
+ \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
+ \end{tikzpicture}\end{center}
+
+ \begin{itemize}
+ \item
+ Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
+ \item
+ If diagonals are equal length, it is a rectangle
+ \item
+ \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
+ \item
+ Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
+ \end{itemize}
+
+ \subsubsection*{Useful vector properties}
+
+ \begin{itemize}
+ \item
+ \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
+ \(k \in \mathbb{R} \setminus \{0\}\)
+ \item
+ If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
+ least one point in common, then they lie on the same straight line
+ \item
+ \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
+ \item
+ \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
+ \end{itemize}
+
+ \subsection*{Linear dependence}
+
+ \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
+ \begin{align*}
+ 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
+ \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
+ \end{align*}
+
+ \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
+ independent if no vector in the set is expressible as a linear
+ combination of other vectors in set, or if they are parallel.
+
+ \subsection*{Three-dimensional vectors}
+
+ Right-hand rule for axes: \(z\) is up or out of page.
+
+ \tdplotsetmaincoords{60}{120}
+ \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
+ vector/.style={-stealth,red,very thick},
+ vector guide/.style={dashed,gray,thick}]
+
+ %standard tikz coordinate definition using x, y, z coords
+ \coordinate (O) at (0,0,0);
+
+ %tikz-3dplot coordinate definition using x, y, z coords
+
+ \pgfmathsetmacro{\ax}{1}
+ \pgfmathsetmacro{\ay}{1}
+ \pgfmathsetmacro{\az}{1}
+
+ \coordinate (P) at (\ax,\ay,\az);
+
+ %draw axes
+ \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
+ \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
+ \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
+
+ %draw a vector from O to P
+ \draw[vector] (O) -- (P);
+
+ %draw guide lines to components
+ \draw[vector guide] (O) -- (\ax,\ay,0);
+ \draw[vector guide] (\ax,\ay,0) -- (P);
+ \draw[vector guide] (P) -- (0,0,\az);
+ \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
+ \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
+ \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
+ \node[tdplot_main_coords,above right]
+ at (\ax,\ay,\az){(\ax, \ay, \az)};
+ \end{tikzpicture}\end{center}
+
+ \subsection*{Parametric vectors}
+
+ Parametric equation of line through point \((x_0, y_0, z_0)\) and
+ parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
+
+ \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
+
+ \section{Circular functions}
+
+ \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
+
+ \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
+ \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
+
+ \subsection*{Reciprocal functions}
+
+ \subsubsection*{Cosecant}
+
+ \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
+
+ \begin{itemize}
+ \item
+ \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
+ \item
+ \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
+ \item
+ \textbf{Turning points} at
+ \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
+ \item
+ \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
+ \end{itemize}