From 0ec0c7cc19557e7de3fed0909142fe8e3160c3bd Mon Sep 17 00:00:00 2001
From: Gregory Hartman <apexcalculus@gmail.com>
Date: Wed, 12 Aug 2026 14:16:06 +0000
Subject: [PATCH] Changes to Chapter 11

---
 ptx/sec_cross_product.ptx       |  6 ++--
 ptx/sec_dot_product.ptx         | 39 +++++++++++------------
 ptx/sec_lines.ptx               | 44 +++++++++++++-------------
 ptx/sec_planes.ptx              | 55 +++++++++++++++++----------------
 ptx/sec_space_coord.ptx         | 43 +++++++++++++-------------
 ptx/sec_vector_intro.ptx        |  8 ++---
 publication/publication-pdf.ptx |  1 +
 7 files changed, 100 insertions(+), 96 deletions(-)

diff --git a/ptx/sec_cross_product.ptx b/ptx/sec_cross_product.ptx
index fe91400a9..dcc3db0d3 100644
--- a/ptx/sec_cross_product.ptx
+++ b/ptx/sec_cross_product.ptx
@@ -394,7 +394,7 @@
       </statement>
     </theorem>
 
-    <aside xml:id="aside-cross-orthogonal" vshift="4">
+    <aside xml:id="aside-cross-orthogonal" vshift="2">
       <title>The cross product and parallel vectors</title>
 
       <p>
@@ -514,7 +514,7 @@
             <idx><h>right hand rule</h><h>of the cross product</h></idx>
       </p>
 
-      <figure xml:id="fig_crossp_rhr" vshift="0">
+      <figure xml:id="fig_crossp_rhr" vshift="3">
         <caption>Illustrating the Right Hand Rule of the cross product</caption>
 
         <!-- START figures/figcrossp_rhr_3D.asy -->
@@ -1116,7 +1116,7 @@
             So the volume of the parallelepiped is 2 cubic units.
           </p>
 
-          <figure xml:id="fig_crossp6" vshift="-6">
+          <figure xml:id="fig_crossp6" vshift="2">
             <caption>A parallelepiped in <xref ref="ex_crossp6"/></caption>
 
             <!-- START figures/figcrossp6_3D.asy -->
diff --git a/ptx/sec_dot_product.ptx b/ptx/sec_dot_product.ptx
index c2ad08285..3ef14d5b0 100644
--- a/ptx/sec_dot_product.ptx
+++ b/ptx/sec_dot_product.ptx
@@ -269,7 +269,7 @@
 
             size(200,200,IgnoreAspect);
             //currentprojection=perspective(7,2,1);
-            currentprojection=orthographic(4,4,2);
+            currentprojection=orthographic(4,4,3);
             defaultrender.merge=true;
 
             // setup and draw the axes
@@ -747,7 +747,7 @@
     </statement>
   </definition>
 
-  <aside vshift="4">
+  <aside vshift="2">
     <p>
       The term <em>perpendicular</em>
       originally referred to lines.
@@ -1018,22 +1018,7 @@
       <p>
         <ol>
           <li>
-            <p>
-              Applying <xref ref="def_orthogonal_projection"/>, we have
-              <md>
-                <mrow>\proj uv \amp = \frac{\dotp uv}{\dotp vv}\vec v</mrow>
-                <mrow>\amp = \frac{-5}{10}\la 3,1\ra</mrow>
-                <mrow>\amp = \la -\frac32,-\frac12\ra</mrow>
-              </md>.
-              Vectors <m>\vec u</m>, <m>\vec v</m> and
-              <m>\proj uv</m> are sketched in <xref ref="fig_dotp4a"/>.
-              Note how the projection is parallel to <m>\vec v</m>;
-              that is, it lies on the same line through the origin as <m>\vec v</m>,
-              although it points in the opposite direction.
-              That is because the angle between <m>\vec u</m> and <m>\vec v</m> is obtuse (<ie/>, greater than <m>90^\circ</m>).
-            </p>
-
-            <figure xml:id="fig_dotp4a" vshift="0">
+            <figure xml:id="fig_dotp4a" vshift="-1">
               <caption>Sketching the three vectors in <xref ref="ex_dotp4_part1" text="local">Part</xref> of <xref ref="ex_dotp4"/></caption>
               <image width="47%">
                 <shortdescription>
@@ -1075,6 +1060,22 @@
                 </latex-image>
               </image>
             </figure>
+
+            <p>
+              Applying <xref ref="def_orthogonal_projection"/>, we have
+              <md>
+                <mrow>\proj uv \amp = \frac{\dotp uv}{\dotp vv}\vec v</mrow>
+                <mrow>\amp = \frac{-5}{10}\la 3,1\ra</mrow>
+                <mrow>\amp = \la -\frac32,-\frac12\ra</mrow>
+              </md>.
+              Vectors <m>\vec u</m>, <m>\vec v</m> and
+              <m>\proj uv</m> are sketched in <xref ref="fig_dotp4a"/>.
+              Note how the projection is parallel to <m>\vec v</m>;
+              that is, it lies on the same line through the origin as <m>\vec v</m>,
+              although it points in the opposite direction.
+              That is because the angle between <m>\vec u</m> and <m>\vec v</m> is obtuse (<ie/>, greater than <m>90^\circ</m>).
+            </p>
+
           </li>
 
           <li>
@@ -1735,7 +1736,7 @@
           A man slides a box along a ramp that rises 3ft over a distance of 15ft by applying 50lb of force as shown in <xref ref="fig_dotp7"/>.
           Compute the work done.
         </p>
-        <figure xml:id="fig_dotp7" vshift="0">
+        <figure xml:id="fig_dotp7" vshift="-2">
           <caption>Computing work when sliding a box up a ramp in <xref ref="ex_dotp7"/></caption>
           <!-- START figures/fig_dotp7.tex -->
           <image width="47%">
diff --git a/ptx/sec_lines.ptx b/ptx/sec_lines.ptx
index 00b13aae2..7a7ec259a 100644
--- a/ptx/sec_lines.ptx
+++ b/ptx/sec_lines.ptx
@@ -813,7 +813,7 @@
           </md>.
         </p>
 
-        <figure xml:id="fig_lines3" vshift="4">
+        <figure xml:id="fig_lines3" vshift="5">
           <caption>Graphing the lines in <xref ref="ex_lines3"/></caption>
 
           <!-- START figures/figlines3_3D.asy -->
@@ -912,7 +912,7 @@
       will help establish a general method of computing this distance <m>h</m>.
     </p>
 
-    <figure xml:id="fig_lines_dist1" vshift="1">
+    <figure xml:id="fig_lines_dist1" vshift="2">
       <caption>Establishing the distance from a point to a line</caption>
       <!-- START figures/fig_lines_dist1.tex -->
       <image width="47%">
@@ -965,26 +965,7 @@
       <video youtube="ZJ_e_0s2s2M" label="vid-vectors-lines-distance-point"/>
     </figure>
 
-    <p>
-      It is also useful to determine the distance between lines,
-      which we define as the length of the shortest line segment that connects the two lines
-      (an argument from geometry shows that this line segments is perpendicular to both lines).
-      Let lines <m>\vec\ell_1(t) = \vec p_1 + t\vec d_1</m> and <m>\vec\ell_2(t) = \vec p_2 + t\vec d_2</m> be given,
-      as shown in <xref ref="fig_lines_dist2"/>.
-      To find the direction orthogonal to both <m>\vec d_1</m> and <m>\vec d_2</m>,
-      we take the cross product:
-      <m>\vec c = \vec d_1\times \vec d_2</m>.
-      The magnitude of the orthogonal projection of
-      <m>\overrightarrow{P_1P_2}</m> onto <m>\vec c</m> is the distance <m>h</m> we seek:
-      <md>
-        <mrow>h\amp =    \norm{\operatorname{proj}_{\vec c}\overrightarrow{P_1P_2}}</mrow>
-        <mrow>\amp = \norm{\frac{\overrightarrow{P_1P_2}\cdot\vec c}{\dotp cc}\vec c}</mrow>
-        <mrow>\amp =\frac{\abs{\overrightarrow{P_1P_2}\cdot \vec c}}{\vnorm c^2}\vnorm c</mrow>
-        <mrow>\amp =\frac{\abs{\overrightarrow{P_1P_2}\cdot \vec c}}{\vnorm c}</mrow>
-      </md>.
-    </p>
-
-    <figure xml:id="fig_lines_dist2" vshift="0.5">
+    <figure xml:id="fig_lines_dist2" vshift="-1">
       <caption>Establishing the distance between lines</caption>
 
       <!-- START figures/figlines_dist2_3D.asy -->
@@ -1094,6 +1075,25 @@
       <!-- figures/figlines_dist2_3D.asy END -->
     </figure>
 
+    <p>
+      It is also useful to determine the distance between lines,
+      which we define as the length of the shortest line segment that connects the two lines
+      (an argument from geometry shows that this line segments is perpendicular to both lines).
+      Let lines <m>\vec\ell_1(t) = \vec p_1 + t\vec d_1</m> and <m>\vec\ell_2(t) = \vec p_2 + t\vec d_2</m> be given,
+      as shown in <xref ref="fig_lines_dist2"/>.
+      To find the direction orthogonal to both <m>\vec d_1</m> and <m>\vec d_2</m>,
+      we take the cross product:
+      <m>\vec c = \vec d_1\times \vec d_2</m>.
+      The magnitude of the orthogonal projection of
+      <m>\overrightarrow{P_1P_2}</m> onto <m>\vec c</m> is the distance <m>h</m> we seek:
+      <md>
+        <mrow>h\amp =    \norm{\operatorname{proj}_{\vec c}\overrightarrow{P_1P_2}}</mrow>
+        <mrow>\amp = \norm{\frac{\overrightarrow{P_1P_2}\cdot\vec c}{\dotp cc}\vec c}</mrow>
+        <mrow>\amp =\frac{\abs{\overrightarrow{P_1P_2}\cdot \vec c}}{\vnorm c^2}\vnorm c</mrow>
+        <mrow>\amp =\frac{\abs{\overrightarrow{P_1P_2}\cdot \vec c}}{\vnorm c}</mrow>
+      </md>.
+    </p>
+
     <p>
       A problem in the Exercise section is to show that this distance is 0 when the lines intersect.
       Note the use of the Triple Scalar Product:
diff --git a/ptx/sec_planes.ptx b/ptx/sec_planes.ptx
index 5fba99a73..1b967adac 100644
--- a/ptx/sec_planes.ptx
+++ b/ptx/sec_planes.ptx
@@ -218,7 +218,7 @@
           The plane is sketched in <xref ref="fig_planes1"/>.
         </p>
 
-        <figure xml:id="fig_planes1" vshift="0">
+        <figure xml:id="fig_planes1" vshift="2">
           <caption>Sketching the plane in <xref ref="ex_planes1"/></caption>
 
           <!-- START figures/figplanes1_3D.asy -->
@@ -743,7 +743,7 @@
           illustrated in <xref ref="fig_planes5"/>.
         </p>
 
-        <figure xml:id="fig_planes5" vshift="0">
+        <figure xml:id="fig_planes5" vshift="4">
           <caption>Illustrating the intersection of a line and a plane in <xref ref="ex_planes5"/></caption>
 
           <!-- START figures/figplanes5_3D.asy -->
@@ -820,31 +820,7 @@
       it is also often necessary to find the distance from a point to a plane.
     </p>
 
-    <p>
-      Consider <xref ref="fig_planes_dist"/>,
-      where a plane with normal vector <m>\vec n</m> is sketched containing a point <m>P</m> and a point <m>Q</m>,
-      not on the plane, is given.
-      We measure the distance from <m>Q</m> to the plane by measuring the length of the projection of <m>\overrightarrow{PQ}</m> onto <m>\vec n</m>.
-      That is, we want:
-      <md number="yes" xml:id="eq_plane_dist">
-        \snorm{\text{ proj } _{\,\vec n}\,{\overrightarrow{PQ}}} = \snorm{\frac{\vec n\cdot \overrightarrow{PQ}}{\vnorm n^2}\vec n} = \frac{\abs{\vec n\cdot \overrightarrow{PQ}}}{\vnorm n}
-      </md>
-    </p>
-
-    <figure xml:id="vid-vectors-planes-distance" component="video" vshift="0">
-      <caption>Video introduction to <xref ref="subsec_planes_distances"/></caption>
-      <video youtube="DT4JMy5Ak54" label="vid-vectors-planes-distance"/>
-    </figure>
-
-    <p>
-      <xref ref="eq_plane_dist">Equation</xref> is important as it does more than just give the distance between a point and a plane.
-      We will see how it allows us to find several other distances as well:
-      the distance between parallel planes and the distance from a line and a plane.
-      Because <xref ref="eq_plane_dist">Equation</xref> is important,
-      we restate it as a Key Idea.
-    </p>
-
-    <figure xml:id="fig_planes_dist" vshift="-4">
+    <figure xml:id="fig_planes_dist" vshift="-1">
       <caption>Illustrating finding the distance from a point to a plane</caption>
 
       <!-- START figures/figplanes_dist_3D.asy -->
@@ -919,6 +895,31 @@
       <!-- figures/figplanes_dist_3D.asy END -->
     </figure>
 
+
+    <p>
+      Consider <xref ref="fig_planes_dist"/>,
+      where a plane with normal vector <m>\vec n</m> is sketched containing a point <m>P</m> and a point <m>Q</m>,
+      not on the plane, is given.
+      We measure the distance from <m>Q</m> to the plane by measuring the length of the projection of <m>\overrightarrow{PQ}</m> onto <m>\vec n</m>.
+      That is, we want:
+      <md number="yes" xml:id="eq_plane_dist">
+        \snorm{\text{ proj } _{\,\vec n}\,{\overrightarrow{PQ}}} = \snorm{\frac{\vec n\cdot \overrightarrow{PQ}}{\vnorm n^2}\vec n} = \frac{\abs{\vec n\cdot \overrightarrow{PQ}}}{\vnorm n}
+      </md>
+    </p>
+
+    <figure xml:id="vid-vectors-planes-distance" component="video" vshift="0">
+      <caption>Video introduction to <xref ref="subsec_planes_distances"/></caption>
+      <video youtube="DT4JMy5Ak54" label="vid-vectors-planes-distance"/>
+    </figure>
+
+    <p>
+      <xref ref="eq_plane_dist">Equation</xref> is important as it does more than just give the distance between a point and a plane.
+      We will see how it allows us to find several other distances as well:
+      the distance between parallel planes and the distance from a line and a plane.
+      Because <xref ref="eq_plane_dist">Equation</xref> is important,
+      we restate it as a Key Idea.
+    </p>
+
     <insight xml:id="idea_planes_dist">
       <title>Distance from a Point to a Plane</title>
       <p>
diff --git a/ptx/sec_space_coord.ptx b/ptx/sec_space_coord.ptx
index 967e484ca..2430240e8 100644
--- a/ptx/sec_space_coord.ptx
+++ b/ptx/sec_space_coord.ptx
@@ -280,7 +280,7 @@
           </md>.
         </p>
 
-        <figure xml:id="fig_space1" vshift="-1">
+        <figure xml:id="fig_space1" vshift="3">
           <caption>Plotting points <m>P</m> and <m>Q</m> in <xref ref="ex_space1"/></caption>
           <!-- START figures/figspace1_3D.asy -->
           <image width="47%">
@@ -1653,19 +1653,10 @@
           The curve is sketched in <xref ref="fig_surfrev1a_3D"/> and the surface is drawn in <xref ref="fig_surfrev1b_3D"/>.
         </p>
 
-        <p>
-          Note how the surface
-          (and hence the resulting equation)
-          is the same if we began with the curve <m>x=\sin(z)</m>,
-          which is also drawn in <xref ref="fig_surfrev1a_3D"/>.
-        </p>
-        <figure xml:id="fig_surfrev1">
-          <caption>Revolving <m>y=\sin(z)</m> about the <m>z</m>-axis in <xref ref="ex_surfrev1"/></caption>
-          <sidebyside widths="47% 47%" margins="0%">
-            <figure xml:id="fig_surfrev1a_3D">
-              <caption/>
+            <figure xml:id="fig_surfrev1a_3D" vshift="4">
+              <caption>Drawing <m>y=\sin(z)</m> and <m>x=\sin(z)</m> in their respective planes to aid the understanding of revolving <m>y=\sin(z)</m> about the <m>z</m>-axis in <xref ref="ex_surfrev1"/></caption>
               <!-- START figures/figsurfrev1a_3D.asy -->
-              <image>
+              <image width="47%">
                 <shortdescription>
                   Graph of two functions y = sin(z) and x = sin(z).
                 </shortdescription>
@@ -1720,12 +1711,19 @@
                 </asymptote>
               </image>
             <!-- figures/figsurfrev1a_3D.asy END -->
-            </figure>
+            </figure>        
+        
+        <p>
+          Note how the surface
+          (and hence the resulting equation)
+          is the same if we began with the curve <m>x=\sin(z)</m>,
+          which is also drawn in <xref ref="fig_surfrev1a_3D"/>.
+        </p>
 
-            <figure xml:id="fig_surfrev1b_3D">
-              <caption/>
+            <figure xml:id="fig_surfrev1b_3D" vshift="-2">
+              <caption>Revolving <m>y=\sin(z)</m> about the <m>z</m>-axis in <xref ref="ex_surfrev1"/></caption>
               <!-- START figures/figsurfrev1b_3D.asy -->
-              <image>
+              <image width="47%">
                 <shortdescription>
                   Graph showing surface formed by revolving y = sin (z) about the z axis.
                 </shortdescription>
@@ -1779,10 +1777,7 @@
                 </asymptote>
               </image>
             <!-- figures/figsurfrev1b_3D.asy END -->
-            </figure>
-          </sidebyside>
-        </figure>
-      </solution>
+            </figure>      </solution>
       <solution component="video" vshift="3">
         <title>Video solution</title>
         <video width="98%" youtube="jrJXSnHU-0o" label="vid-vectors-spacecoord-surfrev-eg" component="video"/>
@@ -2293,6 +2288,8 @@
       <idx><h>quadric surface</h><h>elliptic paraboloid</h></idx>
     </p>
 
+    <p xml:id="pagebreak-elliptic-cone"></p>
+
     <p>
       <dl>
         <li>
@@ -2736,6 +2733,8 @@
       <idx><h>quadric surface</h><h>sphere</h></idx>
     </p>
 
+    <p xml:id="pagebreak-hyperboloid-one-sheet"></p>
+
     <p>
       <dl>
         <li>
@@ -3147,6 +3146,8 @@
       <idx><h>quadric surface</h><h>hyperboloid of two sheets</h></idx>
     </p>
 
+    <p xml:id="pagebreak-hyperbolic-paraboloid"></p>
+
     <p>
       <dl>
         <li>
diff --git a/ptx/sec_vector_intro.ptx b/ptx/sec_vector_intro.ptx
index 86ca0533b..02cd1d4aa 100644
--- a/ptx/sec_vector_intro.ptx
+++ b/ptx/sec_vector_intro.ptx
@@ -546,7 +546,7 @@
         </md>.
       </p>
 
-      <figure xml:id="fig_vect2" vshift="0">
+      <figure xml:id="fig_vect2" vshift="2">
         <caption>Graphing the sum of vectors in <xref ref="ex_vect2"/></caption>
         <!-- START figures/fig_vect2.tex -->
         <image width="47%">
@@ -820,7 +820,7 @@
               </md>.
             </p>
 
-            <figure xml:id="fig_vect3" vshift="0">
+            <figure xml:id="fig_vect3" vshift="2">
               <caption>Graphing vectors <m>\vec v</m> and <m>2\vec v</m> in <xref ref="ex_vect3"/></caption>
             <!-- START figures/fig_vect3.tex -->
               <image width="47%">
@@ -1108,7 +1108,7 @@
               This is sketched in <xref ref="fig_vect5"/>.
             </p>
 
-            <figure xml:id="fig_vect5" vshift="1">
+            <figure xml:id="fig_vect5" vshift="2">
               <caption>Graphing vectors in <xref ref="ex_vect5"/>. All vectors shown have their initial point at the origin</caption>
             <!-- START figures/fig_vect5.tex -->
               <image width="47%">
@@ -1222,7 +1222,7 @@
 
     <p>
       <m>\vec 0</m> is directionless;
-      because <m>\vnorm{0}=0</m>,
+      because <m>\norm{0}=0</m>,
       there is no unit vector in the <q>direction</q> of <m>\vec 0</m>.
     </p>
 
diff --git a/publication/publication-pdf.ptx b/publication/publication-pdf.ptx
index 38fcaff9a..81998aa18 100644
--- a/publication/publication-pdf.ptx
+++ b/publication/publication-pdf.ptx
@@ -47,6 +47,7 @@
       <insertions pagebreaks="exset-double-polar-special exset-cylindrical-spherical-setup exset-Greens-divergence exset-vectors-balance-forces"/>
       <!-- other page breaks -->
       <insertions pagebreaks="pagebreak-taylor-series pagebreak-iterated-int pagebreak-glossary"/>
+      <insertions pagebreaks="pagebreak-elliptic-cone pagebreak-hyperboloid-one-sheet pagebreak-hyperbolic-paraboloid"/>
     </latex>
     <epub>
       <cover front="cover/cover.png"/>