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Commit bf2f398d authored by Tung Anh Vu's avatar Tung Anh Vu
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Initialize repository with content

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*.aux
*.fdb_latexmk
*.fls
*.log
*.nav
*.out
*.pdf
*.snm
*.toc
MAIN=main.tex
all: $(MAIN)
latexmk -latex=xelatex -pdf $<
clean:
latexmk -c $(MAIN)
rm -f *.nav *.snm
find . -name '*-converted-to.pdf' | xargs rm
\input style
\ehyph
\def\proj{\mathop{\rm proj}}
\headline={%
Introductory Review Course for the Bachelor's Programme in Computer Science -- Exercises
\vadjust{\hrule}
}
\begtask
Find all solutions:
$$
\eqalign{
3x + y &= 2 \cr
x^3 + y - 2 &= 0
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
2x^2 - 2x - y &= 14 \cr
2x - y &= -2
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
x^3 - y &= 0 \cr
x - y &= 0
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
y &= x^2 \cr
x^2 + (y - 2)^2 &= 4
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
3x^2 + 2y^2 = 35 \cr
4x^2 - 3y^2 = 24
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
x^2 - xy + y^2 &= 21 \cr
x^2 + 2xy - 8y^2 &= 0
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
4x + y - 3z &= 11 \cr
2x - 3y + 2z &= 9 \cr
x + y + z &= -3
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
3x - 2y + 4z &= 1 \cr
x + y - 2z &= 3 \cr
2x - 3y + 6z &= 8
}
$$
\endtask
\begtask
Find all solutions:
$$
\eqalign{
x - y + 2z &= 2 \cr
x + 2y - z &= 5 \cr
5x - 8y + 13z &= 7
}
$$
\endtask
\begtask
Find the equation of the parabola~$y = ax^2 + bx + c$ that passes through the points~$(2, 0)$, $(3, -1)$, and~$(4, 0)$.
\endtask
\begtask
Find the unit vector in the same direction as~$(-24, -7)$.
\endtask
\begtask
Find a vector with magnitude~3 in the same direction as~$(4, -4)$.
\endtask
\begtask
Find the magnitude of the vector~$(-\sqrt{3}, 3)$.
\endtask
\begtask
Find the dot product of vectors~$(-4, 1)$ and~$(2, -3)$.
\endtask
\begtask
Find a value~$k$ so that vectors~$(2, 4)$ and~$(k, -5)$ are orthogonal.
\endtask
\begtask
Let~$u = (3, 4)$ and~$v = (8, 2)$.
Find the projection of~$u$ onto~$v$.
Then write~$u$ as a sum of two orthogonal vectors with~$\proj_v(u)$ being one of them.
\endtask
\begtask
Find the equation of the circle $x^2 + y^2 + Dx + Ey + F = 0$ that passes through the points~$(-3, -1)$, $(2, 4)$, and~$(-6, 8)$.
\endtask
\begtask
Identify the center and the radius of the circle~$x^2 - 14x + y^2 + 8y + 40 = 0$.
\endtask
\begtask
Find the equation of the tangent line to the circle~$x^2 + y^2 = 25$ at the point~$(3, -4)$.
\endtask
\begtask
Find the center, vertices, foci, and eccentricity of the ellipse~${(x - 4)^2 \over 16} + {(y + 1)^2 \over 25} = 1$.
\endtask
\begtask
Consider the ellipse defined by~$9x^2 + 4y^2 + 36x - 24y + 36 = 0$.
Find the standard form of the equation of the ellipse.
Find the ellipse's center, vertices, foci, and eccentricity.
\endtask
\begtask
Find the distance between points~$(-1, 4, -2)$ and~$(6, 0, 9)$.
\endtask
\begtask
Let~$\vec{u} = (6, 2, 1)$ and~$\vec{v} = (1, 3, -2)$.
Find the cross product~$\vec{u} \times \vec{v}$.
Show that it is orthogonal to both~$\vec{u}$ and~$\vec{v}$.
\endtask
\begtask
Let~$\vec{u} = (1, 1, -1)$, and~$\vec{v} = (1, 1, 1)$.
Find a unit vector that is orthogonal to both~$\vec{u}$ and~$\vec{v}$.
\endtask
\begtask
Let~$\vec{u} = (2, 2, -3)$ and~$\vec{v} = (0, 2, 3)$.
Find the area of the parallelogram that has~$\vec{u}$ and~$\vec{v}$ as adjacent sides.
\endtask
\begtask
Find the area of the triangle with vertices $(0, 0, 0)$, $(1, 2, 3)$, and~$(-3, 0, 0)$.
\endtask
\begtask
Let~$p = (-4, -1, 0)$, and~$\vec{v} = (3, 8, -6)$.
Find a set of parametric equations and a set of symmetric equations for the line that passes through~$p$ and is parallel to~$\vec{v}$.
\endtask
\begtask
Find the general form of the equation of the plane that passes through~$(5, 6, 3)$ and is normal to the vector~$(-2, 1, -2)$.
\endtask
\begtask
Find the general form of the equation of the plane that passes through the points $(2, 3, -2)$, $(3, 4, 2)$ and $(1, -1, 0)$.
\endtask
\begtask
Find a set of parametric equations of the line that passes through $(2, 3, 4)$ and is parallel to the $xz$-plane and the $yz$-plane.
\endtask
\begtask
Consider the planes $3x - 4y + 5z = 6$ and $x + y - z = 2$.
Find parametric equations of their line of intersection.
\endtask
\bye
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Diskrétní matematika -- Cvičení #1\hfill#2, #3
\vadjust{\hrule}
}
}
\def\hwheader#1#2{% cislo domaciho ukolu, termin odevzdani
\headline={%
Diskrétní matematika -- Domácí úkol č. #1\hfill Deadline: #2
\strut\vadjust{\hrule}
}
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% variant, date
\def\testheader#1#2{% cislo kvizu, datum
\vbox{\centerline{\typoscale[2000/] Diskrétní matematika -- kvíz č. #1}\kern3.4pt\hrule}
\ifnum\sharescreenactive = 0
Jméno, příjmení:
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\ifnum\sharescreenactive = 0
Počet odevzdaných listů (vč. tohoto):
\fi
\hfill Datum: #2
\vskip0.4em
\hrule
\vskip0.4em
{\bf Pokyny.}\quad Test vypracujte samostatně, bez jakýchkoli pomůcek (vyjma psacích potřeb).
Nepoužívejte červenou barvu.
Vždy uvádějte postup, zdůvodněte všechna tvrzení.
Kromě případů, kdy se zadání explicitně ptá, znalosti z přednášek nemusíte dokazovat, stačí se na ně odkázat názvem.
Výsledek bez uvedeného postupu má mizivou bodovou hodnotu.
Výsledek jasně označte.
Co je nečitelné nebo nesrozumitelné, nebude hodnoceno.
{\bf Není-li něco jasné, ptejte se!}
\vskip0.4em
\hrule
\vskip1em
}
\footline={\hfil}
\let\algo=\it
\let\defi=\it
\def\mod{\mathop{\rm mod}}
\def\Ham{\Delta_H}
\def\div{\mathrel{\mid}}
\def\abs#1{\mid #1 \mid}
\def\N{{\bbchar{N}}}
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File added
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\documentclass{beamer}
\usetheme{default}
\usepackage{amsmath}
\DeclareMathOperator{\vecproj}{proj}
\usepackage{gensymb}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\projection}[2]{\vecproj_{#2}(#1)}
\title{Systems of equations, Analytic geometry}
\author{Tung Anh Vu}
% TODO: Make a combined logo for KAM as on the department website.
\titlegraphic{%
\includegraphics[width=0.4\textwidth]{logos/kam_logo.eps}
\hspace{2em}
\includegraphics[width=0.5\textwidth]{logos/logotyp_fakulty2.eps}
}
\begin{document}
\begin{frame}
\vfill
\titlepage
\vfill
\usebeamerfont{institute} Contact: \url{tung@kam.mff.cuni.cz}
\end{frame}
\begin{frame}{Systems of equations}
\framesubtitle{One variable, one equation}
Types of equations:
\begin{itemize}
\item \emph{Linear}: \[6x + 3 = 0.\]
\item \emph{Quadratic}: \[2x^2 + 3x + 1 = 0.\]
\item \emph{Cubic}: \[x^3 - 5x^2 - 2x + 24 = 0.\]
\item \emph{Quartic}, \emph{quintic},\dots
\pause
\item Can have 0, 1, multiple, or infinitely many solutions.
\end{itemize}
\end{frame}
\begin{frame}{Solving linear equations}
Linear equations can have either:
\begin{itemize}
\item zero solutions \[7x + 3 = 7x + 2,\]
\item one solution \[6x + 9 = x - 6,\]
\item infinitely many solutions \[5x + 3 - 4x= 3 + x.\]
\end{itemize}
\end{frame}
\begin{frame}{Solving quadratic equations}
\begin{block}{General form}
\[
ax^2 + bx + c = 0,
\]
where \(b, c \in \R\) and \(a \in \R \setminus \{0\}\).
\end{block}
\pause
\begin{block}{Example}
Given \(2x^2 + 3x + 1 = 0\), we have \(a = 2\), \(b = 3\), \(c = 1\).
\end{block}
\end{frame}
\begin{frame}{Solving quadratic equations: quadratic formula}
\begin{block}{Quadratic formula}
\[x_{1, 2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
\end{block}
\pause
\begin{block}{Task}
Solve \(2x^2 + 3x + 1 = 0\) using the quadratic formula.
\end{block}
\end{frame}
\begin{frame}{Solving polynomial equations: by factoring}
\begin{block}{Rational zero test}
Each \textbf{rational solution~\(x\)} of a polynomial equation is of the form~\(\frac{p}{q}\) where
\begin{itemize}
\item \(p\) is a factor of the constant term, and
\item \(q\) is a factor of the leading term.
\end{itemize}
\end{block}
\pause
\begin{block}{Tasks}
Solve the following by factoring:
\begin{itemize}
\item \(x^2 + 2x - 15 = 0\),
\item \(x^3 -7x + 6= 0\).
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Multivariate equations}
\framesubtitle{One equation}
\begin{itemize}
\item Over reals~\(\R\) has generally infinitely many solutions.
\item Over integers~\(\Z\) may be extremely difficult to solve.
\begin{itemize}
\item E.g., Fermat's last theorem.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}{Two equations, two variables}
\framesubtitle{Number of solutions}
\begin{itemize}
\item Can have 0, 1, multiple or infinitely many solutions.
\pause
\item If the equation is linear, then each equation defines a line in~\(\R^2\).
\pause
\item And the solution is the intersection of those lines.
\end{itemize}
\end{frame}
\begin{frame}{Two equations, two variables}
\framesubtitle{Method of substitution}
\begin{enumerate}
\item Solve
\[
\begin{split}
x^2 + 4x - y &= 7 \\
2x - y &= -1
\end{split}
\].
\item Solve
\[
\begin{split}
-x + y &= 4 \\
x^2 + y &= 3
\end{split}
\]
\end{enumerate}
\end{frame}
\begin{frame}{Two equations, two variables}
\framesubtitle{Method of elimination}
\begin{enumerate}
\item Solve
\[
\begin{split}
5x + 3y &= 9 \\
2x - 4y &= 14
\end{split}
\]
\item Solve
\[
\begin{split}
x - 2y &= 3 \\
-2x + 4y &= 1
\end{split}
\]
\item Solve
\[
\begin{split}
2x - y &= 1 \\
4x - 2y &= 2
\end{split}
\]
\end{enumerate}
\end{frame}
\begin{frame}{Analytic geometry}
Study of geometry using a coordinate system.
\end{frame}
\begin{frame}{Vectors}
\textbf{Vector}: geometric object with \emph{direction} and \emph{magnitude}.
\pause
\begin{block}{Example}
Suppose we are in the Euclidean plane~\(\R^2\).
Consider points~\(p = (4, -7)\) and~\(q = (-1, 5)\).
Draw the vector from~\(p\) to~\(q\).
\end{block}
\pause
\begin{block}{Example}
Consider the vector~\(\vec{pq}\) from the previous example.
What is its \emph{angle}?
\end{block}
\end{frame}
\begin{frame}{What can we do with vectors?}
Suppose we have vectors~\(\vec{u}, \vec{v}, \vec{w} \in \R^n\), and real numbers~\(\alpha, \beta \in \R\).
\pause
\begin{itemize}
\item \emph{Addition}: \(\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2 + \cdots + u_n + v_n)\).
\pause
\item \emph{Scalar multiplication}: \(\alpha \vec{u} = (\alpha u_1, \alpha u_2, \ldots, \alpha u_n)\).
\pause
\end{itemize}
Properties of above operations:
\begin{itemize}
\item \emph{Commutativity}: \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\).
\pause
\item \emph{Associativity}: \((\vec{u} + \vec{v}) + \vec{w} = \vec{v} + (\vec{u} + \vec{w})\).
\pause
\item \emph{Distributivity over scalar multiplication}: \((\alpha + \beta)\vec{u} = \alpha\vec{u} + \beta\vec{u}\).
\pause
\item \emph{Distributivity over addition}: \(\alpha(\vec{u} + \vec{v}) = \alpha\vec{u} + \alpha\vec{v}\).
\end{itemize}
\end{frame}
\begin{frame}{Length of a vector}
\begin{block}{Computing the length}
\[
\|\vec{u}\| = \sqrt{u_1^2 + u_2^2 + \cdots + u_n^2}.
\]
\end{block}
\pause
Is it true that~\(\|\alpha \vec{u}\| = \alpha \|\vec{u}\|\)?
\pause
No, but\(\|\alpha \vec{u}\| = |\alpha| \|\vec{u}\|\).
\pause
\begin{block}{Computing the unit vector}
\[
\frac{\vec{u}}{\|\vec{u}\|}.
\]
\end{block}
\end{frame}
\begin{frame}
An airplane is descending at 200 km/hr at an angle of 30 degrees below the horizon.
Find the component form of its velocity vector.
\end{frame}
\begin{frame}{Dot product}
\begin{definition}
Suppose we have~\(\vec{u}, \vec{v}, \vec{w} \in \R^n\).
The \emph{dot product\footnote{You will see during your studies that there are multiple types of dot products. This one is usually known as the \emph{standard dot product}.} of~\(\vec{u}\) and~\(\vec{v}\)} is defined as
\[
\vec{u} \cdot \vec{v} = (u_1 v_1, u_2 v_2, \ldots, u_n v_n).
\]
\end{definition}
\pause
\begin{block}{Properties}
\begin{itemize}
\item \emph{Commutativity}: \(\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}\).
\pause
\item \(\vec{0} \cdot \vec{v} = \vec{0}\).
\pause
\item \emph{Distributivity}: \(\vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w}\).
\pause
\item \(\vec{v} \cdot \vec{v} = \|\vec{v}\|^2\).
\pause
\item \emph{Triangle inequality}: \(\|\vec{u} + \vec{v}\| \le \|\vec{u}\| + \|\vec{v}\|\).
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Dot product in the plane}
Let~\(\vec{u}, \vec{v} \in \R^2\), and~\(\theta\)~be the angle between~\(\vec{u}\) and~\(\vec{v}\). Then
\[
\vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos \theta.
\]
\pause
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
\(\theta\) in degrees & \(\theta\) in radians & \(\vec{u} \cdot \vec{v}\) \\
\hline
90\degree & \(\frac{\pi}{2}\) rad & 0 \\
\hline
0\degree & 0 rad & \(\|\vec{u}\| \|\vec{v}\|\) \\
\hline
180\degree & \(\pi\) rad & \(-\|\vec{u}\| \|\vec{v}\|\) \\
\hline
\end{tabular}
\end{center}
\end{frame}
\begin{frame}{Projection}
\begin{definition}
\emph{Projection} of vector~\(\vec{u}\) on vector~\(v\) is the vector
\[
\projection{u}{v} = \frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|}\cdot \vec{v} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \cdot \vec{v}.
\]
\end{definition}
\end{frame}
\begin{frame}{Circles}
\begin{definition}
A \emph{circle} is a set of {\color{red}{equidistant}} points from a fixed point~\((h, k)\) called the \emph{center}.
The distance from the center to any of the circle points is called the \emph{radius}.
\end{definition}
\pause
\begin{block}{Standard form of the equation of a circle}
\[
(x - h)^2 + (y - k)^2 = r^2
\]
\end{block}
\end{frame}
\begin{frame}
\begin{enumerate}
\item A circle has center~\((2, 3)\) and includes the point~\((1, 4)\).
Find its standard equation.
\pause
\item Find the center and the radius of a circle
\[
x^2 - 6x + y^2 - 2y + 6 = 0.
\]
\end{enumerate}
\end{frame}
\begin{frame}{Ellipses}
\begin{definition}
An \emph{ellipse} is the set of points whose sum of distances from two distinct points called {\color{red} foci} is constant.
\end{definition}
\pause
Some terminology:
\begin{itemize}
\item \emph{Center} is the midpoint of the foci.
\pause
\item \emph{Major axis} is the chord through the foci.
\pause
\item The major axis intersects the ellipse at \emph{vertices}.
\pause
\item \emph{Minor axis} is the chord through the center perpendicular to the major axis.
\pause
\item The minor axis intersects the ellipse at \emph{co-vertices}.
\end{itemize}
\end{frame}
\begin{frame}{Properties of ellipses}
\begin{itemize}
\item Consider an ellipse with center at~\((h, k)\), foci at~\((h \pm c, k)\), vertices at~\((h \pm a, k)\), and co-vertices at~\((h, k \pm b)\).
\pause
\item Sum of distance to foci is~\((a + c) + (a - c) = 2a\).
\pause
\item[\(\Rightarrow\)] Distance from a focal point to a co-vertex is~\(a\).
\pause
\item[\(\Rightarrow\)] \(c^2 = a^2 - b^2\).
\pause
\item \emph{Eccentricity} of an ellipse is defined as~\(\frac{c}{a}\).
\pause
\end{itemize}
\begin{block}{Standard equation}
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
\end{block}
\end{frame}
\begin{frame}
\begin{enumerate}
\item Find the equation of an ellipse with foci at \((0, 1)\) and \((4, 1)\) and major axis of length 6.
\pause
\item Find the center and vertices of an ellipse \(x^2 + 4y^2 + 6x - 8y + 9 = 0\).
\end{enumerate}
\end{frame}
\begin{frame}{Cross product}
Only defined in three dimensional spaces.
\begin{definition}
The \emph{cross product} of~\(\vec{u}, \vec{v} \in \R^3\) is defined as
\[
\vec{u} \times \vec{v} = (u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1).
\]
\end{definition}
\end{frame}
\begin{frame}{Cross product: geometric properties}
Let~\(\vec{u}, \vec{v} \in \R^3\), and~\(\theta\)~be the angle between them.
\pause
\begin{itemize}
\item \(\vec{u} \times \vec{v}\) is orthogonal to both~\(\vec{u}\) and~\(\vec{v}\).
\begin{itemize}
\item The ``orthogonal direction'' is determined by convention.
\end{itemize}
\pause
\item \(\vec{u} \times \vec{v} = \|\vec{u}\| \|\vec{v}\| \sin(\theta) \vec{n}\) where~\(\vec{n}\) is the unit vector orthogonal to~\(\vec{u}\) and~\(\vec{v}\).
\pause
\item \(\|\vec{u} \times \vec{v}\|\) is the area of the parallelogram between~\(\vec{u}\) and~\(\vec{v}\).
\end{itemize}
\end{frame}
\begin{frame}{Lines and planes}
\begin{block}{Parametric equation of a line}
Let~\(t \in \R\) be a parameter.
\[x = x_1 + at; y = y_1 + bt; z = z_1 + bt\]
\end{block}
\begin{block}{Symmetric equation of a line}
\[\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}.\]
\end{block}
\pause
\begin{block}{Exercise}
Find the parametric and the symmetric equation of a line passing through points~\((-2, 1, 0)\) and~\((1, 3, 5)\).
\end{block}
\end{frame}
\begin{frame}{Lines and planes}
\begin{itemize}
\item Consider a plane that passes through the point~\((x_1, y_1, z_1)\) and has a normal vector~\((a, b, c)\).
\pause
\item Then for any point~\((x, y, z)\) in the plane we have
\[
(a, b, c) \cdot (x - x_1, y - y_1, z - z_1) = 0.
\]
\pause
\item[\(\Rightarrow\)] Standard equation of a plane
\[
a(x - x_1) + b(y - y_1) + c(z - z_1) = 0.
\]
\pause
\item General form of the equation of a plane
\[
ax + by + cz + d = 0.
\]
\end{itemize}
\end{frame}
\begin{frame}
\begin{enumerate}
\item Find the general equation of the plane passing through~\((2, 1, 1)\), \((0, 4, 1)\), and~\((-2, 1, 4)\).
\pause
\item Find the intersection of planes~\(x - 2y + z = 0\) and~\(2x + 3y - 2z = 0\).
\end{enumerate}
\end{frame}
\end{document}
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