Vector section re-editing WIP

chapters/linear_algebra_intuitive/vectors.tex

@@ -310,8 +310,9 @@ \subsection{Column representation}
 
 \subsection{Cartesian coordiantes and the standard basis vectors}
 
-The space $\Rs[2]$ can be placed inside a Cartesian coordinate system, such that the origin of all vectors is at the point $\bm{O}=(0,0)$ (see \autoref{fig:2d_vec_cartesian}). It is then very common to use the following two vectors as basis vectors: $\hat{x}$, a vector of norm $1$ pointing in the direction of the $x$-axis, and $\hat{y}$, a vector of norm $1$ pointing in the direction of the $y$-axis.
+The space $\Rs[2]$ can be placed inside a Cartesian coordinate system, such that the origin of all vectors is at the point $\bm{O}=(0,0)$. It is then very common to use the following two vectors as basis vectors: $\xr{\hat{x}}$, a vector of norm $1$ pointing in the direction of the $x$-axis, and $\xb{\hat{y}}$, a vector of norm $1$ pointing in the direction of the $y$-axis:
 
+\colorlet{xbcol}{black!30}
 \begin{center}
   \begin{tikzpicture}[]
     \begin{axis}[
@@ -321,33 +322,47 @@ \subsection{Cartesian coordiantes and the standard basis vectors}
       ymin=-2, ymax=2,
       xtick={-2,-1,1,2},
       ytick={-2,-1,1,2},
+      axis line style={xbcol},
+      tick style={xbcol},
+      ticklabel style={xbcol},
+      xlabel style={xbcol},
+      ylabel style={xbcol},
+      major grid style={black!2},
     ]
     \draw[vector, xred] (0,0) -- (1,0) node[midway, below] {$\hat{x}$};
     \draw[vector, xblue] (0,0) -- (0,1) node[midway, left] {$\hat{y}$};
+    \fill (0,0) circle (0.05);
     \end{axis}
   \end{tikzpicture}
 \end{center}
 
+Using these basis vectors, any vector $\vec{v}$ has components that are equal to the coordinates of the point at it head (see \autoref{fig:vectors_std_basis_coords}).
+
 \begin{figure}
   \centering
   \begin{tikzpicture}[]
     \begin{axis}[
       vector plane,
       width=10cm, height=10cm,
-      xmin=-5, xmax=5,
-      ymin=-5, ymax=5,
+      xmin=-3, xmax=3,
+      ymin=-3, ymax=3,
       xticklabels={},
       yticklabels={},
+      axis line style={black!30},
+      tick style={black!30},
+      grid=both,
+      major grid style={black!10},
+      minor grid style={black!5},
+      minor tick num=4,
     ]
-    \draw[vector, xred] (0,0) -- (2,3);
-    \draw[vector, xblue] (0,0) -- (-1,2);
-    \draw[vector, xgreen] (0,0) -- (-3,0);
-    \draw[vector, xpurple] (0,0) -- (-1,-3);
-    \draw[vector, xorange] (0,0) -- (0,-4);
-    \draw[vector, black] (0,0) -- (1,-1);
-    \fill (0,0) circle (0.1);
+    \draw[vector, xred] (0,0) -- (1,0) node[midway, below] (xh) {$\hat{x}$} node[anchor=north west] {$(1,0)$};
+    \draw[vector, xblue] (0,0) -- (0,1) node[midway, left] (yh) {$\hat{y}$} node[anchor=south east] {$(0,1)$};
+    \draw[vector, xpurple] (0,0) -- (2,1) node[midway, above] {$\vec{v}$} node[anchor=west] {$(2,1)$};
+    \draw[vector, xdarkgreen] (0,0) -- (-2.6,1.2) node[midway, above] {$\vec{u}$} node[anchor=south, xshift=5] {$(-2.6,1.2)$};
+    \draw[vector, xorange] (0,0) -- (-2,-2) node[midway, above] {$\vec{w}$} node[anchor=north east] {$(-2,-2)$};
+    \fill (0,0) circle (0.05);
     \end{axis}
   \end{tikzpicture}
-  \caption{Some $2$-dimensional vectors placed on a $2$-dimensional Cartesian coordinate system.}
-  \label{fig:2d_vec_cartesian}
+  \caption{The standard basis vectors $\xr{\hat{x}}=\colvec{\xr{1};\xb{0}}$ and $\xb{\hat{y}}=\colvec{\xr{0};\xb{1}}$ on a $2$-dimensional Cartesian coordinate system, together with the vectors $\xp{\vec{v}}=\colvec{\xr{2};\xb{1}}$, $\xdg{\vec{u}}=\colvec{\xr{-2.6};\xb{1.2}}$ and $\xo{\vec{w}}=\colvec{\xr{-2};\xb{-2}}$.}
+  \label{fig:vectors_std_basis_coords}
 \end{figure}