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6 changes: 3 additions & 3 deletions ptx/sec_cross_product.ptx
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Expand Up @@ -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>
Expand Down Expand Up @@ -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 -->
Expand Down Expand Up @@ -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 -->
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39 changes: 20 additions & 19 deletions ptx/sec_dot_product.ptx
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Expand Up @@ -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
Expand Down Expand Up @@ -747,7 +747,7 @@
</statement>
</definition>

<aside vshift="4">
<aside vshift="2">
<p>
The term <em>perpendicular</em>
originally referred to lines.
Expand Down Expand Up @@ -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>
Expand Down Expand Up @@ -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>
Expand Down Expand Up @@ -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%">
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44 changes: 22 additions & 22 deletions ptx/sec_lines.ptx
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Expand Up @@ -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 -->
Expand Down Expand Up @@ -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%">
Expand Down Expand Up @@ -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 -->
Expand Down Expand Up @@ -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:
Expand Down
55 changes: 28 additions & 27 deletions ptx/sec_planes.ptx
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Expand Up @@ -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 -->
Expand Down Expand Up @@ -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 -->
Expand Down Expand Up @@ -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 -->
Expand Down Expand Up @@ -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>
Expand Down
43 changes: 22 additions & 21 deletions ptx/sec_space_coord.ptx
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Expand Up @@ -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%">
Expand Down Expand Up @@ -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>
Expand Down Expand Up @@ -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>
Expand Down Expand Up @@ -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"/>
Expand Down Expand Up @@ -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>
Expand Down Expand Up @@ -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>
Expand Down Expand Up @@ -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>
Expand Down
8 changes: 4 additions & 4 deletions ptx/sec_vector_intro.ptx
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Expand Up @@ -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%">
Expand Down Expand Up @@ -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%">
Expand Down Expand Up @@ -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%">
Expand Down Expand Up @@ -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>

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1 change: 1 addition & 0 deletions publication/publication-pdf.ptx
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Expand Up @@ -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"/>
Expand Down