From 66a308abb4b3aed0c37bf7d89e0d556c8e278690 Mon Sep 17 00:00:00 2001 From: William Boyles Date: Thu, 7 Jan 2021 14:21:48 -0500 Subject: [PATCH] Work on multicalc ch 0 and 1 Work relevant to #2 and #29. --- multiCalc/Images/vectorValuedFunctions/TN.PNG | Bin 0 -> 10185 bytes .../Images/vectorValuedFunctions/TNB1.png | Bin 0 -> 5265 bytes .../Images/vectorValuedFunctions/TNB2.png | Bin 0 -> 27183 bytes .../osculatingCircle.png | Bin 0 -> 26428 bytes .../vectorValuedFunctions/osculatingPlane.png | Bin 0 -> 17107 bytes multiCalc/backgroundReview/crossProducts.tex | 49 ++++++++++-------- multiCalc/backgroundReview/dotProducts.tex | 29 ++++++----- multiCalc/backgroundReview/vectors.tex | 45 ++++++++++------ multiCalc/main.pdf | Bin 909336 -> 1130240 bytes .../vectorValuedFunctions/CommonVVFs.tex | 2 +- .../DerivativesOfVVFs.tex | 38 ++++++++------ .../vectorValuedFunctions/IntegralsOfVVFs.tex | 3 +- .../OsculatingPlaneCircleBHat.tex | 30 +++++++---- .../ReparameterizationArcLength.tex | 19 ++++--- multiCalc/vectorValuedFunctions/circles.tex | 8 +-- multiCalc/vectorValuedFunctions/curvature.tex | 7 +-- multiCalc/vectorValuedFunctions/helices.tex | 6 ++- multiCalc/vectorValuedFunctions/lines.tex | 17 +++--- multiCalc/vectorValuedFunctions/nHat.tex | 37 +++++++------ multiCalc/vectorValuedFunctions/planes.tex | 21 +++++--- 20 files changed, 188 insertions(+), 123 deletions(-) create mode 100644 multiCalc/Images/vectorValuedFunctions/TN.PNG create mode 100644 multiCalc/Images/vectorValuedFunctions/TNB1.png create mode 100644 multiCalc/Images/vectorValuedFunctions/TNB2.png create mode 100644 multiCalc/Images/vectorValuedFunctions/osculatingCircle.png create mode 100644 multiCalc/Images/vectorValuedFunctions/osculatingPlane.png diff --git a/multiCalc/Images/vectorValuedFunctions/TN.PNG b/multiCalc/Images/vectorValuedFunctions/TN.PNG new file mode 100644 index 0000000000000000000000000000000000000000..d65fd8bc90f15718b541eb6f31bd6760e16efafd GIT binary patch literal 10185 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literal 0 HcmV?d00001 diff --git a/multiCalc/backgroundReview/crossProducts.tex b/multiCalc/backgroundReview/crossProducts.tex index e6b968f..9802a21 100644 --- a/multiCalc/backgroundReview/crossProducts.tex +++ b/multiCalc/backgroundReview/crossProducts.tex @@ -1,19 +1,21 @@ \subsection{Cross Products} \noindent -A cross product is a way of multiplying two vectors so that the result is a vector. Although the cross product technically only works for 3D vectors, we will first look a a "fake" 2D version to build an intuition. +A cross product is a way of multiplying two vectors so that the result is a vector. +Although the cross product technically only works for 3D vectors, we will first look a a "fake" 2D version to build an intuition. \begin{equation*} \vec{a}\times\vec{b} = a_1b_1-a_2b_2 \end{equation*} -This "fake" 2D cross product gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$. +This ``fake'' 2D cross product gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$. \begin{equation*} \vec{a}\times\vec{b} = \norm{\vec{a}}\norm{\vec{b}}\sin{\theta} \end{equation*} -where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.\\ +where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. Another way to think of the magnitude of the cross product, both in 2D and 3D, is as a measure of how perpendicular two vectors are. \begin{figure}[H] \centering - \includegraphics[scale=0.5]{Images/backgroundReview/CrossProduct} + \includegraphics[width=0.5\textwidth]{./Images/backgroundReview/CrossProduct.png} + \caption{Visualization of the cross product} \end{figure} \noindent @@ -25,38 +27,39 @@ \subsection{Cross Products} \begin{equation*} \norm{\vec{a}\times\vec{b}}=\norm{\vec{a}}\norm{\vec{b}}\sin{\theta} \end{equation*} -where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.\\ -Each component of $\vec{a}\times\vec{b}$ gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$ in some plane: The x-component of $\vec{a}\times\vec{b}$ gives the area in the yz-plane (x = 0 plane).\\ +where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. +Each component of $\vec{a}\times\vec{b}$ gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$ in some plane: +The $x$-component of $\vec{a}\times\vec{b}$ gives the area in the yz-plane ($x$ = 0 plane). $\vec{a}\times\vec{b}$ is perpendicular, also called "normal," to the plane containing $\vec{a}$ and $\vec{b}$. It's direction, is determined by the right hand rule.\\ \noindent -The cross product table of the standard basis vectors is useful for providing some insight into the properties of the cross product. +This cross product table of the standard basis vectors is useful for providing some insight into the properties of the cross product. \begin{table}[H] \centering \renewcommand{\arraystretch}{1.5} - \begin{tabular}{|l|l|l|l|} + \begin{tabular}{|c||c|c|c|} \hline - $\overrightarrow{\text{row}}\times\overrightarrow{\text{col}}$ & $\hat{i}$ & $\hat{j}$ & $\hat{k}$ \\ \hline + $\overrightarrow{\text{row}}\times\overrightarrow{\text{col}}$ & $\hat{i}$ & $\hat{j}$ & $\hat{k}$ \\ \hline\hline $\hat{i}$ & $0$ & $\hat{k}$ & $-\hat{j}$ \\ \hline $\hat{j}$ & $-\hat{k}$ & $0$ & $\hat{i}$ \\ \hline $\hat{k}$ & $\hat{j}$ & $-\hat{i}$ & $0$ \\ \hline \end{tabular} \end{table} -\begin{itemize} - \item \textbf{NOT} Commutative, but is antisymmetric - \begin{equation*} - \vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right) - \end{equation*} - \item Scalar Associative - \begin{equation*} - \left(c\cdot\vec{a}\right)\times\vec{b}=\vec{a}\times\left(c\cdot\vec{b}\right) - \end{equation*} - \item Distributive - \begin{equation*} - \vec{a}\times\left(\vec{b}\times\vec{c}\right) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c} - \end{equation*} -\end{itemize} +\begin{enumerate}[label=] + \item \textbf{\underline{NOT} Commutative}, but is antisymmetric + \begin{equation*} + \vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right) + \end{equation*} + \item \textbf{Scalar Associative} + \begin{equation*} + \left(c\cdot\vec{a}\right)\times\vec{b}=\vec{a}\times\left(c\cdot\vec{b}\right) + \end{equation*} + \item \textbf{Distributive} + \begin{equation*} + \vec{a}\times\left(\vec{b}\times\vec{c}\right) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c} + \end{equation*} +\end{enumerate} \noindent One can also think of the cross product as the determinant of a matrix. diff --git a/multiCalc/backgroundReview/dotProducts.tex b/multiCalc/backgroundReview/dotProducts.tex index 4d1863c..d994b22 100644 --- a/multiCalc/backgroundReview/dotProducts.tex +++ b/multiCalc/backgroundReview/dotProducts.tex @@ -1,32 +1,37 @@ \subsection{Dot Products} \noindent -A dot product is a way of multiplying two vectors so that the result is a scalar. $\vec{a}\cdot\vec{b} = \norm{\vec{a}}\norm{\vec{b}}\cos{\theta}$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. One way to think of the dot product is as a measure of how much two vectors point in the same direction.\\ -We can also show using the law of cosines that $\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+...+a_nb_n$.\\ +A dot product is a way of multiplying two vectors so that the result is a scalar. +$\vec{a}\cdot\vec{b} = \norm{\vec{a}}\norm{\vec{b}}\cos{\theta}$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. +One way to think of the dot product is as a measure of how much two vectors point in the same direction. +We can also show using the law of cosines that $\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+...+a_nb_n$. Knowing the lengths of two vectors and their dot product we can calculate the angle between them as \begin{equation*} - \theta = \arccos{\left(\frac{\vec{a} \cdot \vec{b}}{\norm{\vec{a}} \norm{\vec{b}}}\right)} + \theta = \arccos{\left(\frac{\vec{a} \cdot \vec{b}}{\norm{\vec{a}} \norm{\vec{b}}}\right)}. \end{equation*} \begin{figure}[H] \centering - \includegraphics[scale=0.33]{Images/backgroundReview/DotProduct} + \includegraphics[width=0.25\textwidth]{./Images/backgroundReview/DotProduct.png} + \caption{Two vectors and the angle between them} \end{figure} \noindent Although similar to scalar multiplication, dot products have some properties that set them apart. -\begin{itemize} - \item Commutative +\begin{enumerate}[label=] + \item \textbf{Commutative} \begin{equation*} \vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a} \end{equation*} the same as scalar multiplication. - \item Distributive + \item \textbf{Distributive} \begin{equation*} \vec{a}\cdot\left(\vec{b}+\vec{c}\right) = \vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c} \end{equation*} the same as scalar multiplication. - \item \textbf{NOT} Associative $\left(\vec{a}\cdot\vec{b}\right)\cdot\vec{c}$ is a nonsense expression. However, like scalar multiplication, dot products are scalar associative. - \begin{equation*} - \left(c\cdot\vec{a}\right)\cdot\vec{b} = \vec{a}\cdot\left(c\cdot\vec{b}\right) - \end{equation*} -\end{itemize} \ No newline at end of file + \item \textbf{\underline{NOT} Associative} + $\left(\vec{a}\cdot\vec{b}\right)\cdot\vec{c}$ is a nonsense expression. + However, like scalar multiplication, dot products are scalar associative. + \begin{equation*} + \left(c\cdot\vec{a}\right)\cdot\vec{b} = \vec{a}\cdot\left(c\cdot\vec{b}\right) + \end{equation*} +\end{enumerate} \ No newline at end of file diff --git a/multiCalc/backgroundReview/vectors.tex b/multiCalc/backgroundReview/vectors.tex index fbad189..16e15a2 100644 --- a/multiCalc/backgroundReview/vectors.tex +++ b/multiCalc/backgroundReview/vectors.tex @@ -1,28 +1,35 @@ \section{Vectors} \noindent -A vector is a quantity with both direction and magnitude. One can think of it as a directed line segment. In multivariable calculus, we mostly will work with vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$, but vectors can exist in other dimensions.\\ +A vector is a quantity with both direction and magnitude. +One can think of it as a directed line segment. +In multivariable calculus, we mostly will work with vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$, but vectors can exist in other dimensions.\\ \noindent -Numerical (scalar) quantities have vector analogues, many of which show up in physics. Speed becomes velocity, distance becomes displacement, and mass becomes weight.\\ +Numerical (scalar) quantities have vector analogues, many of which show up in physics. +Speed becomes velocity, distance becomes displacement, and mass becomes weight.\\ \noindent Say we have a 2D vector, $\vec{v} = \langle v_x, v_y \rangle$. \begin{figure}[H] \centering - \includegraphics[scale=0.5]{Images/backgroundReview/VectorAddition} + \includegraphics[scale=0.5]{./Images/backgroundReview/VectorAddition.png} + \caption{The $x$ and $y$ components of a vector $v$} \end{figure} \noindent -Its length, also called magnitude or norm, is notated $\norm{\vec{v}} = \sqrt{v_x^2+v_y^2}$. This pattern of the norm being equal to the square-root of the sum of the squares of the vector's components continues into higher dimensions.\\ +Its length, also called magnitude or norm, is notated $\norm{\vec{v}} = \sqrt{v_x^2+v_y^2}$. +This pattern of the norm being equal to the square-root of the sum of the squares of the vector's components continues into higher dimensions.\\ \noindent -The angle a 2D vector forms with the horizontal axis is $\theta = \tan^{-1}{\left(\frac{v_y}{v_x}\right)}$. There is not a useful version of this formula in higher dimensions.\\ +The angle a 2D vector forms with the horizontal axis is $\theta = \tan^{-1}{\left(\frac{v_y}{v_x}\right)}$. +There is not a useful version of this formula in higher dimensions. Using $\theta$ and $\norm{\vec{v}}$, we can see that $v_x = \norm{\vec{v}}\cos{\theta}$ and $v_y = \norm{\vec{v}}\sin{\theta}$.\\ \noindent -Vectors can be added and subtracted from each other in a way that the result is another vector. We do this numerically by adding the corresponding components of each vector.\\ -For example, if $\vec{a} = \langle 1,3 \rangle$ and $\vec{b} = \langle 4,7 \rangle$, then $\vec{a}+\vec{b} = \langle 1+4, 3+7 \rangle = \langle 5,10 \rangle$.\\ +Vectors can be added and subtracted from each other in a way that the result is another vector. +We do this numerically by adding the corresponding components of each vector. +For example, if $\vec{a} = \langle 1,3 \rangle$ and $\vec{b} = \langle 4,7 \rangle$, then $\vec{a}+\vec{b} = \langle 1+4, 3+7 \rangle = \langle 5,10 \rangle$ and $\vec{b}-\vec{a} = \langle 4-1, 7-3, \rangle = \langle 3,4 \rangle$. \noindent @@ -30,7 +37,8 @@ \section{Vectors} \begin{figure}[H] \centering - \includegraphics[scale=0.33]{Images/backgroundReview/Parallelogram} + \includegraphics[scale=0.33]{./Images/backgroundReview/Parallelogram.png} + \caption{Visualization of $\vec{v}+\vec{w}$ and $\vec{v} - \vec{w}$} \end{figure} \noindent @@ -38,27 +46,31 @@ \section{Vectors} \begin{figure}[H] \centering - \includegraphics[scale=0.5]{Images/backgroundReview/ScalarMultiples} + \includegraphics[scale=0.5]{./Images/backgroundReview/ScalarMultiples.png} + \caption{A vector $\vec{v}$ scaled by different constants} \end{figure} \noindent -A unit vector is any vector with magnitude 1. Rather than using an arrow like for other vectors, unit vectors are notated with a carat $\left(\wedge \right)$ over top, like $\hat{i}$, which is read as "i hat".\\ -We an transform any vector with non-zero magnitude into a unit vector by dividing the vector by its norm. This normalized vector will point in the same direction as the original vector.\\ +A unit vector is any vector with magnitude 1. +Rather than using an arrow like for other vectors, unit vectors are notated with a carat $\left(\wedge \right)$ over top, like $\hat{i}$, which is read as ``i hat''. +We can transform any vector with non-zero magnitude into a unit vector by dividing the vector by its norm. +This normalized vector will point in the same direction as the original vector.\\ \noindent -It is common in mathematics for $\hat{i} = \langle 1,0,0 \rangle$ to be the unit vector in the x-direction, $\hat{j} = \langle 0,1,0 \rangle$ to be the unit vector in the y-direction, and $\hat{k} = \langle 0,0,1 \rangle$ to be the unit vector in the z-direction.\\ -Together, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are called the standard basis vectors because all other vectors can be written as linear combination of these. +It is common in mathematics for $\hat{i} = \langle 1,0,0 \rangle$ to be the unit vector in the x-direction, $\hat{j} = \langle 0,1,0 \rangle$ to be the unit vector in the y-direction, and $\hat{k} = \langle 0,0,1 \rangle$ to be the unit vector in the z-direction. +Together, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are called the standard basis vectors because all other vectors in $\R^3$ can be written as linear combination of these. \begin{figure}[H] \centering - \includegraphics[scale=0.33]{Images/backgroundReview/UnitVectors} + \includegraphics[scale=0.33]{./Images/backgroundReview/UnitVectors.png} + \caption{The standard basis vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$} \end{figure} \input{./backgroundReview/dotProducts} \input{./backgroundReview/crossProducts} \noindent -Now that we have defined the dot product and cross product, we can put the two together as the scalar triple product, which gives the volume of the parallelepiped spanned by $\vec{a}$, $\vec{b}$, and $\vec{c}$.\\ +Now that we have defined the dot product and cross product, we can put the two together as the scalar triple product, which gives the volume of the parallelepiped spanned by $\vec{a}$, $\vec{b}$, and $\vec{c}$. \begin{equation*} \vec{a}\cdot\left(\vec{b}\times\vec{c}\right) = \det\begin{bmatrix} a_1 & a_2 & a_2 \\ @@ -69,5 +81,6 @@ \section{Vectors} \begin{figure}[H] \centering - \includegraphics[scale=0.33]{Images/backgroundReview/Parallelipiped} + \includegraphics[width=0.5\textwidth]{./Images/backgroundReview/Parallelipiped.png} + \caption{Scalar triple product gives volume of parallelepiped spanned by three vectors.} \end{figure} diff --git a/multiCalc/main.pdf b/multiCalc/main.pdf index f416e710873988907698b0fdf992fb5ed4060c93..7091fcf15cd21101142701f360c96022d8f2add9 100644 GIT binary patch literal 1130240 zcmd3P2Uru^7OuUZf*lo=2o^*I(i57EW~GBjlM;%QKteG{QL#7lAYehTqoQI*rKwm! zL8U1mh=?eP2q=hB-kQmz0TJFc@B7}l=es#1nLTS({a4vLGmLcf=HXSyQZruUKYT4k z!l~hyj%%efHF2t@I8}Y7855_fgHv*HbMWzFuvFA$WTdAPEe 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d0eece3..d10b319 100644 --- a/multiCalc/vectorValuedFunctions/CommonVVFs.tex +++ b/multiCalc/vectorValuedFunctions/CommonVVFs.tex @@ -8,7 +8,7 @@ \section{Common Vector-Valued Functions} \noindent For example, if $\vec{r}(t) = \langle \tan{t},6t,\ln{\left(16-t^2\right)} \rangle$, \begin{itemize} - \item $\tan{t}$ is defined for all real numbers equivalent to $\pm\pi/2$ radians. + \item $\tan{t}$ is defined for all real numbers not equivalent to $\pm\pi/2$ radians. \item $6t$ is defined for all real numbers. \item $\ln{\left(16-t^2\right)}$ is defined for $t\in\left(-4,4\right)$. \end{itemize} diff --git a/multiCalc/vectorValuedFunctions/DerivativesOfVVFs.tex b/multiCalc/vectorValuedFunctions/DerivativesOfVVFs.tex index 07a1fa7..c9158e1 100644 --- a/multiCalc/vectorValuedFunctions/DerivativesOfVVFs.tex +++ b/multiCalc/vectorValuedFunctions/DerivativesOfVVFs.tex @@ -1,58 +1,64 @@ \section{Derivatives of VVFs} \noindent -Just like functions from Calc I and II, we can differentiate VVFs. In fact, the limit definitions of the derivative are nearly identical.\\ +Just like functions from Calc I and II, we can differentiate VVFs. +In fact, the limit definitions of the derivative are nearly identical. Let $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$. \begin{align*} \vec{r^\prime}(t) &= \lim_{h\to 0}{\frac{\vec{r}(t+h)-\vec{r}(t)}{h}} \\ - &= \lim_{h\to 0}{\langle \frac{x(t+h)-x(t)}{h}, \frac{y(t+h)-y(t)}{h}, \frac{z(t+h)-z(t)}{h} \rangle} + &= \lim_{h\to 0}{\bigg\langle \frac{x(t+h)-x(t)}{h}, \frac{y(t+h)-y(t)}{h}, \frac{z(t+h)-z(t)}{h} \bigg\rangle} \end{align*} The limit distributes inside the vector, so \begin{equation*} - \vec{r^\prime}(t) = \langle x^{\prime}(t), y^{\prime}(t), z^{\prime}(t) \rangle + \vec{r^\prime}(t) = \langle x^{\prime}(t), y^{\prime}(t), z^{\prime}(t) \rangle. \end{equation*} \noindent -Like a position function from Calc I and II, the derivative of a VVF representing position gives a VVF representing velocity, and the 2nd derivative gives a VVF representing acceleration. The magnitude of the velocity VVF, the speed, is commonly notated $v(t)$.\\ +Like a position function from Calc I and II, the derivative of a VVF representing position gives a VVF representing velocity, and the 2nd derivative gives a VVF representing acceleration. +The magnitude of the velocity VVF, the speed, is commonly notated $v(t)$.\\ \noindent -There are 5 important properties of the derivatives of VVFs. These properties are similar to single-variable derivatives.\\ -Let $\vec{r}(t)$ and $\vec{s}(t)$ be VVFs, $a(t)$ be a scalar function, and $c$ be a scalar.\\ -\begin{enumerate} - \item Linearity +There are 5 important properties of the derivatives of VVFs. +These properties are similar to single-variable derivatives. +Let $\vec{r}(t)$ and $\vec{s}(t)$ be VVFs, $a(t)$ be a scalar function, and $c$ be a scalar. +\begin{enumerate}[label=] + \item \textbf{Linearity} \begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}c\vec{r}(t) = c\vec{r^\prime}(t) \end{equation*} - \item Product Rule for Scalar Functions + \item \textbf{Product Rule for Scalar Functions} \begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}a(t)\vec{r}(t) = a(t)\vec{r^\prime}(t) + \vec{r}(t)a^{\prime}(t) \end{equation*} - \item Dot Product Rule + \item \textbf{Dot Product Rul}e \begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}\vec{s}(t)\cdot\vec{r}(t) = \vec{s}(t)\cdot\vec{r^\prime}(t) + \vec{r}(t)\vec{s^\prime}(t) \end{equation*} - \item Cross Product Rule + \item \textbf{Cross Product Rule} \begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}\vec{s}(t)\times\vec{r}(t) = \vec{s}(t)\times\vec{r^\prime}(t) + \vec{s^\prime}(t)\times\vec{r}(t) \end{equation*} - \item Chain Rule + \item \textbf{Chain Rule} \begin{equation*} \frac{\mathrm{d}}{\mathrm{d}t}\vec{r}(a(t)) = \vec{r^\prime}(a(t))a^{\prime}(t) \end{equation*} \end{enumerate} -A quotient rule doesn't make sense because we don't have an operation for dividing two vectors by each other. +A quotient rule doesn't make sense because we don't have an operation for dividing two vectors by each other.\\ \noindent -Just like in single variable calculus, we can use the derivative of VVFs to find tangent lines to the curve. Similar to how $f^{\prime}(a)$ represents the slope of $f$ at $a$, $\vec{r^\prime}(a)$ represents the direction of the tangent line at $a$. Remembering the VVF form of a line, the tangent line to $\vec{r}$ at $t$ is +Just like in single variable calculus, we can use the derivative of VVFs to find tangent lines to the curve. +Similar to how $f^{\prime}(a)$ represents the slope of $f$ at $a$, $\vec{r^\prime}(a)$ represents the direction of the tangent line at $a$. +Remembering the VVF form of a line, the tangent line to $\vec{r}$ at $t$ is \begin{equation*} - \vec{l}(t)=\vec{r}(t)+t\vec{r^\prime}(t) + \vec{l}(t)=\vec{r}(t)+t\vec{r^\prime}(t). \end{equation*} In fact, tangent lines appear so often, that we have a special unit vector representing the direction of the tangent line. \begin{equation*} \hat{T}(t) = \frac{\vec{r^\prime}(t)}{\norm{\vec{r^\prime}(t)}} \end{equation*} -You can remember $\hat{T}$ as the "tangent" vector. +You can remember $\hat{T}$ as the ``tangent'' vector. \begin{figure}[H] \centering \includegraphics[scale=0.33]{Images/vectorValuedFunctions/TangentVector} + \caption{$\hat{T}$ is $\vec{r^\prime}$ normalized.} \end{figure} \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/IntegralsOfVVFs.tex b/multiCalc/vectorValuedFunctions/IntegralsOfVVFs.tex index 4e618f0..607c084 100644 --- a/multiCalc/vectorValuedFunctions/IntegralsOfVVFs.tex +++ b/multiCalc/vectorValuedFunctions/IntegralsOfVVFs.tex @@ -1,5 +1,6 @@ \section{Integrals of VVFs} -VVFs can also be integrated. The integration operation also distributes inside the vector, for both definite and indefinite integrals. +VVFs can also be integrated. +The integration operation also distributes inside the vector, for both definite and indefinite integrals. \begin{equation*} \int{\vec{r}(t)\mathrm{d}t} = \left<\int{x(t)\mathrm{d}t},\int{y(t)\mathrm{d}t},\int{z(t)\mathrm{d}t}\right> \end{equation*} diff --git a/multiCalc/vectorValuedFunctions/OsculatingPlaneCircleBHat.tex b/multiCalc/vectorValuedFunctions/OsculatingPlaneCircleBHat.tex index 1f0df0f..6e6fe63 100644 --- a/multiCalc/vectorValuedFunctions/OsculatingPlaneCircleBHat.tex +++ b/multiCalc/vectorValuedFunctions/OsculatingPlaneCircleBHat.tex @@ -1,23 +1,35 @@ \subsection{Osculating Plane/Circle \& B-Hat $\left(\hat{B}\right)$} \noindent -The osculating plane in the plane containing $\vec{r}(t)$, $\hat{T}$ and $\hat{N}$. It is only defined when $\hat{N}\neq 0$. This means strait lines do not have an osculating plane.\\ +The osculating plane in the plane containing $\vec{r}(t)$, $\hat{T}$ and $\hat{N}$. It is only defined when $\hat{N}\neq 0$. This means straight lines do not have an osculating plane. +The osculating circle lives in the osculating plane, is centered at $\vec{r}(t) + \frac{\hat{N}(t)}{\kappa(t)}$, and has radius $\frac{1}{\kappa(t)}$. +The tangent line at $\vec{r}(t)$ is also tangent to the osculating circle because both points have the same curvature. -\noindent -The osculating circle lives in the osculating plane, is centered at $\vec{r}(t) + \frac{\hat{N}(t)}{\kappa(t)}$, and has radius $\frac{1}{\kappa(t)}$. The tangent line at $\vec{r}(t)$ is also tangent to the osculating circle because both points have the same curvature. - -[INSERT IMAGES] +\begin{figure}[H] + \label{osculating_plane_circle} + \centering + \includegraphics[width = 0.33\textwidth]{./Images/vectorValuedFunctions/osculatingCircle.png} + \includegraphics[width = 0.33\textwidth]{./Images/vectorValuedFunctions/osculatingPlane.png} + \caption{Osculating circle and plane} +\end{figure} \noindent The vector that is normal to the plane is \begin{equation*} - \hat{B}(t) = \hat{T}\times\hat{N} + \hat{B}(t) = \hat{T}\times\hat{N}, \end{equation*} -which is called the binormal vector because it is perpendicular to both $\hat{T}$ and $\hat{N}$. Together, $\hat{T}$, $\hat{N}$, and $\hat{B}$ form the Frenet Serret Frame, also called the TNB frame. +which is called the ``binormal'' vector because it is perpendicular to both $\hat{T}$ and $\hat{N}$. +Together, $\hat{T}$, $\hat{N}$, and $\hat{B}$ form the Frenet Serret Frame, also called the TNB frame. -[INSERT IMAGES] +\begin{figure}[H] + \label{osculating_plane_circle} + \centering + \includegraphics[width = 0.33\textwidth]{./Images/vectorValuedFunctions/TNB1.png} + \includegraphics[width = 0.5\textwidth]{./Images/vectorValuedFunctions/TNB2.png} + \caption{TNB frame} +\end{figure} \noindent We can write the equation for the osculating plane as \begin{equation*} - \hat{B}(t)\cdot\left(\langle x,y,z\rangle - \vec{r}(t)\right) = 0 + \hat{B}(t)\cdot\left(\langle x,y,z\rangle - \vec{r}(t)\right) = 0. \end{equation*} \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/ReparameterizationArcLength.tex b/multiCalc/vectorValuedFunctions/ReparameterizationArcLength.tex index 5796769..0bd5cc0 100644 --- a/multiCalc/vectorValuedFunctions/ReparameterizationArcLength.tex +++ b/multiCalc/vectorValuedFunctions/ReparameterizationArcLength.tex @@ -1,27 +1,30 @@ \section{Reparameterization \& Arc Length} \noindent -VVFs can be reparameterized to trace out the same curve at different speeds by replacing $t$ in $\vec{r}(t)$ with any non-decreasing function of $t$. This fact can come in handy to make the bounds of an integration problem more convenient.\\ +VVFs can be reparameterized to trace out the same curve at different speeds by replacing $t$ in $\vec{r}(t)$ with any non-decreasing function of $t$. +This fact can come in handy to make the bounds of an integration problem more convenient.\\ \noindent The integral of the derivative of a VVF gives the displacement vector because \begin{equation*} - \int_{a}^{b}{\vec{r^\prime}(t)\mathrm{d}t}=\vec{r}(b)-\vec{r}(a) + \int_{a}^{b}{\vec{r^\prime}(t)\mathrm{d}t}=\vec{r}(b)-\vec{r}(a). \end{equation*} -This is exactly like how velocity $\cdot$ time $=$ displacement. +This is exactly like how $\text{veclvity} \cdot \text{time} = \text{displacment}$. \noindent -If we integrate the magnitude of $\vec{r^\prime}9t)$, we can use the fact that $\text{distance} = \text{speed}\cdot\text{time}$ to find the arc length of $\vec{r}(t)$ as +If we integrate the magnitude of $\vec{r^\prime}(t)$, we can use the fact that $\text{distance} = \text{speed} \cdot \text{time}$ to find the arc length of $\vec{r}(t)$ as \begin{equation*} - s=\int{\norm{\vec{r^\prime}(t)}\mathrm{d}t}=\int{\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)^2}\mathrm{d}t} + s=\int{\norm{\vec{r^\prime}(t)}\mathrm{d}t}=\int{\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)^2}\mathrm{d}t}. \end{equation*} We can also write this as an arclength function, \begin{equation*} - s(t)=\int_{0}^{t}{\norm{\vec{r^{\prime}}(\tau)}\mathrm{d}\tau} + s(t)=\int_{0}^{t}{\norm{\vec{r^{\prime}}(\tau)}\mathrm{d}\tau}. \end{equation*} \noindent If we have a function \begin{equation*} - f(t)=s(t)=\int_{0}^{t}{\norm{\vec{r^{\prime}}(\tau)}\mathrm{d}\tau} + f(t)=s(t)=\int_{0}^{t}{\norm{\vec{r^{\prime}}(\tau)}\mathrm{d}\tau}, \end{equation*} -where $s$ is strictly increasing, then $f$ has an inverse by the horizontal line test. That is $t(s) = f^{-1}(s)$ exists and is also non-decreasing. If we reparameterize $\vec{r}(t)$ to $\vec{r}(t(s))$, which is called the arc length parameterization, the parameterization will have a constant speed. \ No newline at end of file +where $s$ is strictly increasing, then $f$ has an inverse by the horizontal line test. +That is, $t(s) = f^{-1}(s)$ exists and is also non-decreasing. +If we reparameterize $\vec{r}(t)$ to $\vec{r}(t(s))$, which is called the arc length parameterization, the parameterization will have a constant speed. \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/circles.tex b/multiCalc/vectorValuedFunctions/circles.tex index 02fb078..42c12ca 100644 --- a/multiCalc/vectorValuedFunctions/circles.tex +++ b/multiCalc/vectorValuedFunctions/circles.tex @@ -1,5 +1,7 @@ \subsection{Circles} \noindent -You should already recognize $x^2 + y^2 = r^2$ as the equation of a circle with radius $r$ centered at the origin. A circle as a VVF is $\vec{r}(t) = \langle r\cos{t}, r\sin{t} \rangle$, which is identical to the parametric form of a circle.\\ -In $\mathbb{R}^3$, the z-component is some constant that tells us which plane, $z=c$, the circle is in.\\ -We can also have circles parallel to $x=0$ and $y=0$ planes by changing the positions of the $\sin$, $\cos$, and $c$ terms. For example, $\vec{r}(t) = \langle \cos{t}, c, \sin{t} \rangle$ is a circle in the $y=c$ plane. \ No newline at end of file +You should already recognize $x^2 + y^2 = r^2$ as the equation of a circle with radius $r$ centered at the origin. +A circle as a VVF in $\R^2$ is $\vec{r}(t) = \langle r\cos{t}, r\sin{t} \rangle$, which is identical to the parametric form of a circle. +In $\mathbb{R}^3$, the z-component is some constant that tells us which plane, $z=c$, the circle is in. +We can also have circles parallel to $x=0$ and $y=0$ planes by changing the positions of the $\sin$, $\cos$, and $c$ terms. +For example, $\vec{r}(t) = \langle \cos{t}, c, \sin{t} \rangle$ is a circle in the $y=c$ plane. \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/curvature.tex b/multiCalc/vectorValuedFunctions/curvature.tex index 9742fea..ba7dc19 100644 --- a/multiCalc/vectorValuedFunctions/curvature.tex +++ b/multiCalc/vectorValuedFunctions/curvature.tex @@ -1,8 +1,9 @@ \subsection{Curvature} -Curvature is 1 divided by the radius of the circle that best approximates the curve at a point. Tighter turns have smaller radii and higher curvature.\\ -We can use $\hat{T}$ to find the curvature at a point on $\vec{r}(t)$.\\ +Curvature is 1 divided by the radius of the circle that best approximates the curve at a point. +Tighter turns have smaller radii and higher curvature. +We can use $\hat{T}$ to find the curvature at a point on $\vec{r}(t)$. \begin{equation*} - \kappa(t) = \norm{\frac{\mathrm{d}\hat{T}}{\mathrm{d}s}} = \norm{\frac{\mathrm{d}\hat{T}}{\mathrm{d}t}\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{-1}} = \norm{\frac{\mathrm{d}\hat{T}}{\mathrm{d}t}}\frac{1}{v(t)} + \kappa(t) = \norm{\dd{\hat{T}}{s}} = \norm{\dd{\hat{T}}{t}\left(\dd{s}{t}\right)^{-1}} = \norm{\dd{\hat{T}}{t}}\frac{1}{v(t)} \end{equation*} \noindent diff --git a/multiCalc/vectorValuedFunctions/helices.tex b/multiCalc/vectorValuedFunctions/helices.tex index 91408e9..bc15828 100644 --- a/multiCalc/vectorValuedFunctions/helices.tex +++ b/multiCalc/vectorValuedFunctions/helices.tex @@ -1,8 +1,10 @@ \subsection{Helices} \noindent -A helix looks like a spring and appears to look like a circle when viewed from top-down. It has the form $\vec{r}(t) = \langle r\cos{t}, r\sin{t}, ct\rangle$ where $a\in\mathbb{R}$. $a$ defines the "tightness" between consecutive windings. +A helix looks like a spring and appears to look like a circle when viewed from the top looking down. +It has the form $\vec{r}(t) = \langle r\cos{t}, r\sin{t}, ct\rangle$ where $a\in\mathbb{R}$. $a$ defines the ``tightness'' between consecutive windings. \begin{figure}[H] \centering - \includegraphics[scale=0.33]{Images/vectorValuedFunctions/Helix} + \includegraphics[width=0.5\textwidth]{./Images/vectorValuedFunctions/Helix.png} + \caption{A helix} \end{figure} \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/lines.tex b/multiCalc/vectorValuedFunctions/lines.tex index 7d25897..c1c4d30 100644 --- a/multiCalc/vectorValuedFunctions/lines.tex +++ b/multiCalc/vectorValuedFunctions/lines.tex @@ -1,17 +1,20 @@ \subsection{Lines} \noindent -A straight line is probably the simplest 3D VVF. We can form any straight line using a point that the line passes through and the direction vector of the line.\\ -Letting $P$ be the point and $\vec{v}$ be the direction, a straight line has the form $\vec{r}(t) = \vec{P}+t\vec{v}$, where $\vec{P}$ is the vector with components the same as $P$. The function's output starts at $\vec{P}$ when $t=0$ and moves in the direction of $\vec{v}$ as $t$ increases. +A straight line is probably the simplest 3D VVF. We can form any straight line using a point that the line passes through and the direction vector of the line. +Letting $P$ be the point and $\vec{v}$ be the direction, a straight line has the form $\vec{r}(t) = \vec{P}+t\vec{v}$, where $\vec{P}$ is the vector with components the same as $P$. +The function's output starts at $\vec{P}$ when $t=0$ and moves in the direction of $\vec{v}$ as $t$ increases. -% It'd be nice if we could get this image to fit on the above page \begin{figure}[H] \centering - \includegraphics[scale=0.33]{Images/vectorValuedFunctions/VectorLine} + \includegraphics[width=0.45\textwidth]{./Images/vectorValuedFunctions/VectorLine.png} + \caption{A line can be formed by a single point and a direction vector or two points.} \end{figure} \noindent -A line can also be formed using two points. To find the equation of the line in this case, let $\vec{v}$ be the vector connecting the two points, and let $\vec{P}$ be a vector point from the origin to one of the two points. We now have an origin point and direction and can write our function as +A line can also be formed using two points. +To find the equation of the line in this case, let $\vec{v}$ be the vector connecting the two points, and let $\vec{P}$ be a vector point from the origin to one of the two points. +We now have an origin point and direction and can write our function as \begin{equation*} - \vec{r}(t) = \vec{P_0} + t\left(\vec{P_1} - \vec{P_0}\right) + \vec{r}(t) = \vec{P_0} + t\left(\vec{P_1} - \vec{P_0}\right), \end{equation*} - Where $P_0$ and $P_1$ are the two points on the line. \ No newline at end of file +where $P_0$ and $P_1$ are the two points on the line. \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/nHat.tex b/multiCalc/vectorValuedFunctions/nHat.tex index eed893a..94e43bb 100644 --- a/multiCalc/vectorValuedFunctions/nHat.tex +++ b/multiCalc/vectorValuedFunctions/nHat.tex @@ -1,28 +1,34 @@ \subsection{N-Hat $\left(\hat{N}\right)$} -\noindent +Starting with the definition of $\hat{T}(t)$, \begin{align*} + \hat{T}(t) &= \frac{\vec{r^\prime}(t)}{v(t)} \\ \vec{r^{\prime}}(t) &= v(t)\hat{T}(t) \\ - \vec{r^{\prime\prime}}(t) &= v(t)\hat{T}^{\prime}(t)+\hat{T}(t)v^{\prime}(t) + \vec{r^{\prime\prime}}(t) &= v(t)\hat{T}^{\prime}(t)+\hat{T}(t)v^{\prime}(t). \end{align*} \noindent We will show that $\hat{T}(t) \perp \hat{T}^{\prime}(t)$. \begin{align*} - \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\hat{T}(t)\cdot\hat{T}(t)\right) &= \hat{T}\cdot\hat{T}^{\prime}(t) \\ - \frac{\mathrm{d}}{\mathrm{d}t}\left(\hat{T}\cdot\hat{T}\right) &= \frac{\mathrm{d}}{\mathrm{d}t}1 = 0 + \frac{1}{2}\dd{}{t}\left(\hat{T}(t)\cdot\hat{T}(t)\right) &= \hat{T}\cdot\hat{T}^{\prime}(t) \\ + \dd{}{t}\left(\hat{T}\cdot\hat{T}\right) &= \dd{}{t}1 = 0 +\end{align*} +So, +\begin{align*} + \hat{T}\cdot\hat{T}^{\prime}(t) &= 0 \implies \hat{T}(t) \perp \hat{T}^{\prime}(t) \\ + \hat{N}(t) &= \frac{\hat{T}^{\prime}(t)}{\norm{\hat{T}^{\prime}(t)}}\perp\hat{T}. \end{align*} -So, -\begin{equation*} - \hat{T}\cdot\hat{T}^{\prime}(t) = 0 \text{ and } \hat{T}(t) \perp \hat{T}^{\prime}(t) -\end{equation*} -\begin{equation*} - \hat{N}(t)=\frac{\hat{T}^{\prime}(t)}{\norm{\hat{T}^{\prime}(t)}}\perp\hat{T} -\end{equation*} \noindent -$\hat{N}$ is a unit vector perpendicular to $\hat{T}$ that points in the direction that the curve curls into. It is called the normal vector because it is perpendicular to the curve. It is in the same plane as $\vec{r}^\prime$, $\hat{T}$, and $\vec{r}^{\prime\prime}$. +$\hat{N}$ is a unit vector perpendicular to $\hat{T}$ that points in the direction that the curve curls into. +It is called the normal vector because it is perpendicular to the curve. +It is in the same plane as $\vec{r}^\prime$, $\hat{T}$, and $\vec{r}^{\prime\prime}$. -[INSERT IMAGE] +\begin{figure}[H] + \label{that_nhat} + \centering + \includegraphics[width = 0.5\textwidth]{./Images/vectorValuedFunctions/TN.PNG} + \caption{$\hat{T}$ and $\hat{N}$} +\end{figure} \noindent $\hat{N}$ allows us to rewrite $\vec{r}^{\prime\prime}$.\\ @@ -31,5 +37,6 @@ \subsection{N-Hat $\left(\hat{N}\right)$} \end{equation*} \noindent -We can see that $\vec{r}^{\prime\prime}(t)$ has two parts. If $\vec{r}(t)$ represents position, then $\frac{\mathrm{d}v}{\mathrm{d}t}$ represents linear acceleration and $v^2(t)\kappa(t)$ represents centripetal acceleration.\\ -You might recognize the formula for centripetal acceleration in the 2nd part. If we let $R(t) = \frac{1}{\kappa(t)}$, then the 2nd part becomes $\frac{v^2(t)}{R(t)}$, which looks exactly like the formula for centripetal acceleration for uniform circular motion: $a_c = \frac{v^2}{r}$. \ No newline at end of file +We can see that $\vec{r}^{\prime\prime}(t)$ has two parts. If $\vec{r}(t)$ represents position, then $\frac{\mathrm{d}v}{\mathrm{d}t}$ represents linear acceleration and $v^2(t)\kappa(t)$ represents centripetal acceleration. +You might recognize the formula for centripetal acceleration in the 2nd part from physics. +If we let $R(t) = \frac{1}{\kappa(t)}$, then the 2nd part becomes $\frac{v^2(t)}{R(t)}$, which looks exactly like the formula for centripetal acceleration for uniform circular motion: $a_c = \frac{v^2}{r}$. \ No newline at end of file diff --git a/multiCalc/vectorValuedFunctions/planes.tex b/multiCalc/vectorValuedFunctions/planes.tex index 149b83b..21b3249 100644 --- a/multiCalc/vectorValuedFunctions/planes.tex +++ b/multiCalc/vectorValuedFunctions/planes.tex @@ -1,30 +1,37 @@ \subsection{Planes} \noindent -A plane can also be formed using a point in the plane, $P$, and a vector perpendicular to the plane, $\vec{n}$. All vectors $\langle x,y,z \rangle$ that originate from $P$ and remain in the plane must be perpendicular to $\vec{n}$, so their dot product with $\vec{n}$ would be 0. So, the point-normal form of a plane is\footnote{Note: Conventionally, $\vec{n}$ is a unit vector, $\hat{n}$.} +A plane can also be formed using a point in the plane, $P$, and a vector perpendicular to the plane, $\vec{n}$. +All vectors $\langle x,y,z \rangle$ that originate from $P$ and remain in the plane must be perpendicular to $\vec{n}$, so their dot product with $\vec{n}$ would be 0. +So, the point-normal form of a plane is\footnote{Conventionally, $\vec{n}$ is a unit vector, $\hat{n}$.} \begin{equation*} - \vec{n}\cdot\left(\langle x,y,z \rangle - \vec{P}\right) = 0 + \vec{n}\cdot\left(\langle x,y,z \rangle - \vec{P}\right) = 0. \end{equation*} \begin{figure}[H] \centering - \includegraphics[scale=0.5]{Images/vectorValuedFunctions/PlaneNormalVector} + \includegraphics[width=0.5\textwidth]{./Images/vectorValuedFunctions/PlaneNormalVector.png} + \caption{A point $P_0$ and a normal vector $\vec{n}$ define a plane.} \end{figure} \noindent -One can also construct a plane from 3 non-collinear points in the plane. One can still take advantage of point-normal form here by choosing 1 point to be $P_0$ and drawing vectors from this point to the two other points. The cross product of these two vectors is $\vec{n}$. +One can also construct a plane from 3 non-collinear points in the plane. +One can still take advantage of point-normal form here by choosing 1 point to be $P_0$ and drawing vectors from this point to the two other points. +The cross product of these two vectors is $\vec{n}$. \begin{equation*} - \left(\left(\vec{P_1} - \vec{P_0}\right) \times \left(\vec{P_2} - \vec{P_0}\right)\right) \cdot \left(\langle x,y,z \rangle - \vec{P_0}\right) = 0 + \left(\left(\vec{P_1} - \vec{P_0}\right) \times \left(\vec{P_2} - \vec{P_0}\right)\right) \cdot \left(\langle x,y,z \rangle - \vec{P_0}\right) = 0, \end{equation*} where $P_0$, $P_1$, and $P_2$ are the three points in the plane.\\ \noindent -One can also construct a plane from a point in the plane, $P_0$, and a line in the plane, $\vec{r}(t) = \vec{P_1} + t\vec{v}$, that doesn't pass through $P_0$. One can get this setup into point-normal form by choosing a an output of $\vec{r}(t)$, like $\vec{P_1}$, and constructing a vector that points from $\vec{P_1}$ to $\vec{P_0}$, $\vec{P_1}-\vec{P_0}$, and crossing this with $\vec{v}$ to find $\vec{n}$. +One can also construct a plane from a point in the plane, $P_0$, and a line in the plane, $\vec{r}(t) = \vec{P_1} + t\vec{v}$, that doesn't pass through $P_0$. +One can get this setup into point-normal form by choosing a an output of $\vec{r}(t)$, like $\vec{P_1}$, and constructing a vector that points from $\vec{P_1}$ to $\vec{P_0}$, $\vec{P_1}-\vec{P_0}$, and crossing this with $\vec{v}$ to find $\vec{n}$. \begin{equation*} \left(\vec{v} \times \left(\vec{P_1} - \vec{P_0}\right)\right) \cdot \left(\langle x,y,z \rangle - \vec{P_0}\right) = 0 \end{equation*} \noindent -One can also construct a plane from two intersecting lines, $\vec{r_1}(t) = \vec{P_0} + t\vec{v_1}$ and $\vec{r_2}(t) = \vec{P_0} + t\vec{v_2}$, where $P_0$ is where the two lines intersect. One can cross $\vec{v_1}$ with $\vec{v_2}$ to get the normal vector. +One can also construct a plane from two intersecting lines, $\vec{r_1}(t) = \vec{P_0} + t\vec{v_1}$ and $\vec{r_2}(t) = \vec{P_0} + t\vec{v_2}$, where $P_0$ is where the two lines intersect. +One can cross $\vec{v_1}$ with $\vec{v_2}$ to get the normal vector. \begin{equation*} \left(\vec{v_1} \times \vec{v_2}\right) \cdot \left(\langle x,y,z \rangle - \vec{P}_{0}\right) = 0 \end{equation*} \ No newline at end of file