diff --git a/source/linear-algebra/source/02-EV/02.ptx b/source/linear-algebra/source/02-EV/02.ptx index c7df14c21..09eb13de3 100644 --- a/source/linear-algebra/source/02-EV/02.ptx +++ b/source/linear-algebra/source/02-EV/02.ptx @@ -50,6 +50,9 @@ since \IR^1=\setBuilder{cx}{c\in\IR}. \draw (0,-0.2) -- (0,0.2) node[above] {0}; \end{tikzpicture} + The real line, with a blue vector pointing to the right beginning at 0 and ending at x, + representing an arbitrary vector in \IR^1. + @@ -69,6 +72,8 @@ since \IR^1=\setBuilder{cx}{c\in\IR}. \draw[<->] (0,-4) -- (0,4); \end{tikzpicture} + The x-y plane. + The xy-plane..
    @@ -116,6 +121,8 @@ since \IR^1=\setBuilder{cx}{c\in\IR}. \draw[->] (0,0,0) -- (0,0,6); \end{tikzpicture} + The coordinate axes representing three dimensional space. + Coordinate axes representing \IR^3.
      @@ -173,6 +180,10 @@ since \IR^1=\setBuilder{cx}{c\in\IR}. \draw[thick,red,->] (0,0,0) -- (-2,0,1); \end{tikzpicture} +

      Two images side by side. The left image is the xy-plane, with a single vector pointing down and to the right. + Its spanning set, the line parallel to it, is also shown.

      +

      The right image illustrates two non-parallel vectors in three dimensional space, as well as the two-dimensional plane they span.

      +       diff --git a/source/linear-algebra/source/02-EV/04.ptx b/source/linear-algebra/source/02-EV/04.ptx index 5c9498625..c38203bf3 100644 --- a/source/linear-algebra/source/02-EV/04.ptx +++ b/source/linear-algebra/source/02-EV/04.ptx @@ -97,6 +97,7 @@ \draw[thick,purple,->] (0,0,0) -- (1,1,-1); \end{tikzpicture} + Three vectors in \IR^3, no two of which are parallel. They all lie in the same plane, which is shown.

      diff --git a/source/linear-algebra/source/03-AT/01.ptx b/source/linear-algebra/source/03-AT/01.ptx index 9c1a27957..90eaf54e9 100644 --- a/source/linear-algebra/source/03-AT/01.ptx +++ b/source/linear-algebra/source/03-AT/01.ptx @@ -105,6 +105,11 @@ Given a linear transformation T:V\to W, \end{scope} \end{tikzpicture} + + The domain \IR^3 is represented on the left by the xyz coordinate axes, along with an arbitrary vector \vec{v}. + A curved dotted arrow to the right points to the co-domain, \IR^2, represented by the xy coordinate axes, along with + an arbitrary vector labeled T(\vec{v}). + @@ -118,7 +123,13 @@ as is necessary for computer animation in film or video games.

      A projection of a 3D teapot onto a 2D screen - + + +

      A computer generated image of a three dimensional teapot sitting in front of a screen that shows a + flattened, two dimensional image of the same teapot. Several parallel black arrows point from identifiable points on the three + dimensional teapot (such as the spout and handle) to the corresponding places on the two dimensional image.

      +       +
      diff --git a/source/linear-algebra/source/03-AT/03.ptx b/source/linear-algebra/source/03-AT/03.ptx index c6e5ed791..214afe9e1 100644 --- a/source/linear-algebra/source/03-AT/03.ptx +++ b/source/linear-algebra/source/03-AT/03.ptx @@ -110,6 +110,12 @@ is an important subspace of V defined by \end{scope} \end{tikzpicture} + Two Euclidean spaces are shown, connected by a curved dashed arrow to the right. + The space on the left is \IR^3, represented by the xyz coordinate axes. A blue + line representing a one dimensional subspace is shown and labeled \ker T. The space on the + right is \IR^2, represented by the xy-plane. The zero vector \vec{0} is shown + and labeled. + @@ -307,6 +313,17 @@ right example's image is a planar subspace of \IR^3. \end{scope} \end{tikzpicture} + +

      Two examples are shown. The one on the left illustrates a transformation from \IR^3 to \IR^2 + with the xyz coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xy-plane. The images of the individual arbitrary vectors are shown, + and the entire xy plane is shaded, representing that they span the entire space \IR^2.

      +

      The example on the right illustrates a transformation from \IR^2to \IR^3 + with the xy coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xyz coordinate axes. The images of the individual arbitrary vectors are + shown, all lying in a two dimensional plane which is shaded. +

      +       diff --git a/source/linear-algebra/source/03-AT/04.ptx b/source/linear-algebra/source/03-AT/04.ptx index 85c929358..356e3d983 100644 --- a/source/linear-algebra/source/03-AT/04.ptx +++ b/source/linear-algebra/source/03-AT/04.ptx @@ -88,6 +88,18 @@ distinct vectors to the same place. More precisely, T is injective if \node[anchor=north] at (5,-1) {not injective}; \end{tikzpicture} + +

      Two examples are shown. The one on the left illustrates a transformation from \IR^2 to \IR^3 + with the xy coordinate axes (with two distinct vectors labeled \vec{v} and \vec{w} shown) + connected by a curved dashed arrow to the right to an image of the xyz coordinate axes. The images of + the individual vectors are shown and are distinct, labeled T(\vec{v}) and T(\vec{w}). This example + is labeled below as "injective".

      +

      The example on the right illustrates a transformation from \IR^3to \IR^3 + with the xyz coordinate axes (with two distinct vectors labeled \vec{v} and \vec{w} shown) + connected by a curved dashed arrow to the right to an image of the xy coordinate axes. The images of + the individual vectors are shown and are the same, labeled T(\vec{v})=T(\vec{w}). This example + is labeled below as "not injective".

      +       @@ -252,6 +264,18 @@ Let T: V \rightarrow W be a linear transformation. \node[anchor=north] at (5,-2) {not surjective}; \end{tikzpicture} + +

      Two examples are shown. The one on the left illustrates a transformation from \IR^3 to \IR^2 + with the xyz coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xy-plane. The images of the individual arbitrary vectors are shown, + and the entire xy plane is shaded, representing that they span the entire space \IR^2. This example is + labeled below as "surjective".

      +

      The example on the right illustrates a transformation from \IR^2to \IR^3 + with the xy coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xyz coordinate axes. The images of the individual arbitrary vectors are + shown, all lying in a two dimensional plane which is shaded. This example is labeled below as "not surjective". +

      +       @@ -435,6 +459,12 @@ recognized by its trivial kernel. \end{scope} \end{tikzpicture} + + The xy-plane on the left, with the zero vector \vec{0} and two distinct, non-parallel arbitrary vectors + labeled \vec{v} and \vec{w}, connected by a curved dashed arrow to the right to the xyz coordinate axes, + representing \IR^3. Three vectors are shown in \IR^3: the zero vector, labeled T(\vec{0})=\vec{0}; + and nonzero, non-parallel vectors T(\vec{v}) and T(\vec{w}). + @@ -537,6 +567,19 @@ recognized by its identical codomain and image. \node[anchor=north] at (5,-2) {not surjective, \(\Im T\not=\IR^3\)}; \end{tikzpicture} + +

      Two examples are shown. The one on the left illustrates a transformation from \IR^3 to \IR^2 + with the xyz coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xy-plane. The images of the individual arbitrary vectors are shown, + and the entire xy plane is shaded, representing that they span the entire space \IR^2. This example is + labeled below as "surjective, \Im T = \IR^2".

      +

      The example on the right illustrates a transformation from \IR^2to \IR^3 + with the xy coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xyz coordinate axes. The images of the individual arbitrary vectors are + shown, all lying in a two dimensional plane which is shaded. This example is labeled below as "not surjective, + \Im T \neq \IR^3". +

      +       @@ -756,6 +799,18 @@ dimension than its codomain, and is therefore not surjective. \node[anchor=north] at (5,-2) {not surjective, \(2<3\)}; \end{tikzpicture} + +

      Two examples are shown. The one on the left illustrates a transformation from \IR^3 to \IR^2 + with the xyz coordinate axes (with two non-parallel vectors \vec{v} and \vec{w} shown) connected by a curved dashed + arrow to the right to an image of the xy-plane. A single vector is shown in the xy-plane, labeled + T(\vec{v})=T(\vec{w}). This example is labeled below as "not injective, 3>2".

      +

      The example on the right illustrates a transformation from \IR^2to \IR^3 + with the xy coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed + arrow to the right to an image of the xyz coordinate axes. The images of the individual arbitrary vectors are + shown, all lying in a two dimensional plane which is shaded. This example is labeled below as "not surjective, + 2 <3". +

      +

      diff --git a/source/linear-algebra/source/04-MX/01.ptx b/source/linear-algebra/source/04-MX/01.ptx index b63025e78..1a410101e 100644 --- a/source/linear-algebra/source/04-MX/01.ptx +++ b/source/linear-algebra/source/04-MX/01.ptx @@ -50,6 +50,12 @@ is a linear map from \IR^n \rightarrow \IR^k. \IR^n \arrow[rr, bend right, "S\circ T"'] \arrow[r,"T"] \& \IR^m \arrow[r,"S"] \&\IR^k \end{tikzcd} + + A representation of the composition of maps. The chain \IR^n \rightarrow \IR^m \rightarrow \IR^k is + adorned with a T labeling the arrow from \IR^n to \IR^m, and a S labeling the arrow + from \IR^m \rightarrow \IR^k. Below this is a curved arrow connecting \IR^n on the left to \IR^k + on the right, which is labeled S \circ T. + The composition of two linear maps. diff --git a/source/linear-algebra/source/05-GT/01.ptx b/source/linear-algebra/source/05-GT/01.ptx index ab533b493..5a09445e0 100644 --- a/source/linear-algebra/source/05-GT/01.ptx +++ b/source/linear-algebra/source/05-GT/01.ptx @@ -76,6 +76,14 @@ transforms the unit square. \draw[red,dashed] (1,0) -- (1,1) -- (0,1); \end{tikzpicture} + + Two vectors are shown in the xy-plane: + A blue vector a long the x-axis is labeled A\vec{e}_1 =\left[\begin{array}{c}2 \\ 0 \end{array}\right] and + a blue vector a long the y-axis is labeled A\vec{e}_2 =\left[\begin{array}{c}0 \\ 3 \end{array}\right]. + Dashed blue lines extend vertically and horizontally from the ends of these vectors to illustrate a rectangle formed + with those two vectors as two of the sides. Additionally, a shaded red unit square is similarly illustrated by the + red vectors \left[\begin{array}{c}1 \\ 0\end{array}\right] and \left[\begin{array}{c}0 \\ 1\end{array}\right]. + Transformation of the unit square by the matrix A. @@ -97,7 +105,7 @@ The image below illustrates how the linear transformation standard matrix B = \left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right] transforms the unit square.

      -
      +
      \begin{tikzpicture} @@ -112,6 +120,14 @@ transforms the unit square. \draw[red,dashed] (1,0) -- (1,1) -- (0,1); \end{tikzpicture} + + Two vectors are shown in the xy-plane: + A blue vector a long the x-axis is labeled B\vec{e}_1 =\left[\begin{array}{c}2 \\ 0 \end{array}\right] and + a blue vector extending upwards to the right is labeled B\vec{e}_2 =\left[\begin{array}{c}3 \\ 4 \end{array}\right]. + Dashed blue lines extend from the ends of these vectors parallel to the other vector to illustrate a parallelogram formed + with those two vectors as two of the sides. Additionally, a shaded red unit square is similarly illustrated by the + red vectors \left[\begin{array}{c}1 \\ 0\end{array}\right] and \left[\begin{array}{c}0 \\ 1\end{array}\right]. + Transformation of the unit square by the matrix B
      @@ -146,7 +162,7 @@ What is the area of the transformed unit square? = 4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right] -
      +
      \begin{tikzpicture} @@ -161,6 +177,14 @@ What is the area of the transformed unit square? \draw[blue,dashed] (2,0) -- (5,2) -- (3,2); \end{tikzpicture} + + Two vectors are shown in the xy-plane: + A blue vector a long the x-axis is labeled B\left[\begin{array}{c}1 \\ 0 \end{array}\right] =2\left[\begin{array}{c}1 \\ 0 \end{array}\right] and + a blue vector extending to the right and upwards is labeled B\left[\begin{array}{c}\frac{3}{4} \\ \frac{1}{2} \end{array}\right] =4\left[\begin{array}{c}\frac{3}{4} \\ \frac{1}{2} \end{array}\right]. + Dashed blue lines extend from the ends of these vectors parallel to the other vector to illustrate a parallelogram formed + with those two vectors as two of the sides. Additionally, a shaded red parallelogram is similarly illustrated by the + red vectors \left[\begin{array}{c}1 \\ 0\end{array}\right] and \left[\begin{array}{c}\frac{3}{4} \\ \frac{1}{2}\end{array}\right]. + Certain vectors are stretched out without being rotated.
      @@ -206,6 +230,9 @@ What is the area of the transformed unit square? \draw[blue,dashed] (2,0) -- (5,2) -- (3,2); \end{tikzpicture} + + The images from and are shown side by side. + A linear map transforming parallelograms into parallelograms.
      @@ -250,6 +277,9 @@ In order to figure out how to compute it, we first figure out the properties it \draw[blue,dashed] (2,0) -- (5,2) -- (3,2); \end{tikzpicture} + + The images from and are shown side by side. + The linear transformation B scaling areas by a constant factor, which we call the determinant
      @@ -277,7 +307,13 @@ area of resulting parallelogram, what is the value of \det([\vec{e}_1\hspace{ \draw[dashed,blue] (1,0) -- (1,1); \draw[dashed,blue] (0,1) -- (1,1); \end{tikzpicture} - + + Two vectors are shown in the xy-plane: + A blue vector a long the x-axis is labeled \vec{e}_1=\left[\begin{array}{c}1 \\ 0 \end{array}\right] and + a blue vector extending along the y-axis is labeled \vec{e}_2 =\left[\begin{array}{c}0 \\ 1 \end{array}\right]. + Dashed blue lines extend from the ends of these vectors parallel to the other vector to illustrate a square formed + with those two vectors as two of the sides. This square is shaded red. + The transformation of the unit square by the identity matrix. @@ -485,6 +521,14 @@ the same parallelogram, but the second matrix reflects its orientation. \draw[red,dashed] (1,0) -- (1,1) -- (0,1); \end{tikzpicture} + + Two images are shown. Both show the xy-plane along with two blue vectors (one extending right, and one extending + upwards and to the right), with dotted lines completing the parallelogram they span. Both images also contain the red unit + square. The image on the left labels the blue vectors as + A\vec{e}_1=\left[\begin{array}{c}2 \\ 0 \end{array}\right] and A\vec{e}_2=\left[\begin{array}{c}3 \\ 4 \end{array}\right], + while the image on the right labels the two vectors as + B\vec{e}_2=\left[\begin{array}{c}2 \\ 0 \end{array}\right] and B\vec{e}_1=\left[\begin{array}{c}3 \\ 4 \end{array}\right]. + Reflection of a parallelogram as a result of swapping columns. @@ -584,6 +628,11 @@ may be verified by adding and subtracting columns. \end{scope} \end{tikzpicture} + + A red square, a larger purple parallelogram, and an even larger blue rectangle are arranged from left to right horizontally. + A curved arrow labeled B points from the red square to the purple parallelogram, and a curved arrow labeled A + points from the purple parallelogram to the blue rectangle. + Area changing under the composition of two linear maps diff --git a/source/linear-algebra/source/05-GT/02.ptx b/source/linear-algebra/source/05-GT/02.ptx index aab0da421..262436fe5 100644 --- a/source/linear-algebra/source/05-GT/02.ptx +++ b/source/linear-algebra/source/05-GT/02.ptx @@ -79,6 +79,16 @@ into the determinant of a smaller matrix. \draw[purple,dashed,very thick] (0,0,0) -- node[left] {\tiny\(h=1\)} (0,1,0); \end{tikzpicture} + + Three vectors in \IR^3 are shown in red: + \left[\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right], + \left[\begin{array}{c}1 \\ 3 \\ 0 \end{array}\right], and + \left[\begin{array}{c}0 \\ 1 \\ 1 \end{array}\right]. + The parallelogram formed by \left[\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right] and + \left[\begin{array}{c}1 \\ 3 \\ 0 \end{array}\right], lying in the xy-plane, is shaded red. + The parallelepiped formed by the three vectors is shown in blue. + Additionally, a dashed red line segment along the z axis spanning the apparent height of the parallelepiped is labeled h=1. + Transformation of the unit cube by the linear transformation. diff --git a/source/linear-algebra/source/05-GT/03.ptx b/source/linear-algebra/source/05-GT/03.ptx index 1a8efede2..8892b10b1 100644 --- a/source/linear-algebra/source/05-GT/03.ptx +++ b/source/linear-algebra/source/05-GT/03.ptx @@ -82,7 +82,7 @@ Consider the linear transformation A : \IR^2 \rightarrow \IR^2 given by the matrix A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right].

      -
      +
      \begin{tikzpicture} @@ -98,6 +98,16 @@ given by the matrix A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{a \draw[purple!50!red,thick,->] (0,0) -- (2,1); \end{tikzpicture} + + A figure in the xy-plane. + A blue vector pointing right along the x-axis is labeled A\vec{e}_1. + A blue vector pointing up and right is labeled A\vec{e}_2. + Dotted blue lines complete the parallelogram spanned by these two vectors. + A red horizontal vector labeled \vec{e}_1 and a red vertical vector labeled \vec{e}_2 + form the edges of a red square. + A red vector points up and to the right through the middle of the parallelogram, + and a purple, longer vector parallel to this vector extends out further. + Transformation of the unit square by the linear transformation A
      @@ -151,6 +161,12 @@ is a vector \vec{x} \in \IR^n such that A\vec{x} is parallel to + + The image from is reproduced with most of the parts grayed out. + The red vector pointing up and to the right is now labeled \left[\begin{array}{c}2 \\ 1 \end{array}\right], + and the purple parallel vector remains; it is labeled to the right with A\left[\begin{array}{c}2 \\ 1\end{array}\right]=3\left[\begin{array}{c}2 \\ 1\end{array}\right]. + The red vector \vec{e}_1 remains, and the parallel blue vector is now labeled A\vec{e}_1=2\vec{e}_1. + The map A stretches out the eigenvector \left[\begin{array}{c}2 \\ 1 \end{array}\right] by a factor of 3 (the corresponding eigenvalue).
      diff --git a/source/linear-algebra/source/applications/pagerank.ptx b/source/linear-algebra/source/applications/pagerank.ptx index d4262fe8a..24ebccf87 100644 --- a/source/linear-algebra/source/applications/pagerank.ptx +++ b/source/linear-algebra/source/applications/pagerank.ptx @@ -15,7 +15,7 @@ The $2,110,000,000,000 Problem In the picture below, each circle represents a webpage, and each arrow represents a link from one page to another.

      -
      +
      \begin{tikzpicture} @@ -45,6 +45,40 @@ represents a link from one page to another. \end{scope} \end{tikzpicture} + +

      A network consisting of seven numbered circles connected by red arrows. The circles are arranged in two rows: the top row + from left to right is circles 1, 4, 5, and 6. The bottom row, from left to right, is circles 2, 3, and 7. Circles 2, 3, and 7 are + directly below circles 1,4, and 5 respectively; no circle is below circle 6.

      +

      + Circle 1 has arrows pointing right to circle 4 and down to circle 2. + An incoming arrow arrives upward from circle 2. +

      +

      + Circle 2 has two arrows pointing out, one back up to circle 1, and one to the right to circle 3. + Arrows come in from circle 1 above, from circle 4 which is above and to the right, and from + circle 7 which is to the right past circle 3. +

      +

      + Circle 3 has a single arrow coming in from the left from circle 2. + Two arrows go outward: one up to circle 4, and one to the right to circle 7. +

      +

      + Circle 4 has arrows come in from circle 1 on the left, circle 3 below, and circle 7 below and right. + A single arrow points out, down and to the left to circle 2. +

      +

      + Circle 5 has one arrow arriving: from the right from circle 6. + Two arrows point out: one to the right to circle 6 and one down to circle 7. +

      +

      + Circle 6 has one arrow coming in from the left from circle 5. + Two arrows point out: to the left to circle 5 and down and left to circle 7. +

      +

      + Circle 7 has two arrows pointing out: left to circle 2 and up and left to circle 4. + Three arrows point in: from the left from circle 3, from above from circle 5, and from above and right from circle 6. +

      +       A seven-webpage network
      @@ -85,7 +119,7 @@ A webpage distributes its importance equally among all the pages it links to

      Consider this small network with only three pages. Let x_1, x_2, x_3 be the importance of the three pages respectively.

      -
      +
      \begin{tikzpicture} @@ -102,6 +136,13 @@ Consider this small network with only three pages. Let x_1, x_2, x_3 be \end{scope} \end{tikzpicture} + +

      + A network consisting of three numbered circles connected by 4 red arrows. + Arrows point from circle 1 to both circle 2 and circle 3. + An arrow points from circle 2 to circle 1, and the final arrow points from circle 3 to circle 2. +

      +       A three-webpage network
      @@ -143,6 +184,9 @@ This corresponds to the page rank system: \end{scope} \end{tikzpicture} + + The image from is reproduced. + A three-webpage network
      @@ -232,6 +276,9 @@ That is, find the eigenspace associated with \lambda=1 for the matrix \end{scope} \end{tikzpicture} + + The image from is reproduced. + A three-webpage network
      @@ -314,6 +361,9 @@ and both pages are twice as important as page 3. \end{scope} \end{tikzpicture} + + The network from is reproduced. + A seven-webpage network @@ -373,6 +423,9 @@ A=\left[\begin{array}{ccccccc} \end{scope} \end{tikzpicture} + + The network from is reproduced. + A seven-webpage network @@ -431,6 +484,9 @@ here is a complete ranking of all seven pages from most important to least impor \end{scope} \end{tikzpicture} + + The network from is reproduced. + A seven-webpage network @@ -476,6 +532,18 @@ from most important to least important. \end{scope} \end{tikzpicture} + +

      A different network with seven numbered circles connected by red arrows.

      +
        +
      • Circle 1 has arrows outward to circles 2 and 5, and inward from circle 3.
        • +
      • Circle 2 has arrows outward to circles 5 and 7, and inward from circles 1,3,6, and 7.
        • +
      • Circle 3 has arrows outward to circles 1 and 2, and a single arrow inward from circle 4.
        • +
      • Circle 4 has a single arrow outward to circle 3, and a single arrow inward from circle 7.
        • +
      • Circle 5 has a single arrow outward to circle 6, and arrows inward from circles 1,2, and 6.
        • +
      • Circle 6 has arrows outward to circles 5 and 2, and a single arrow inward from circle 6.
        • +
      • Circle 7 has arrows outward to circles 2 and 4, and a single arrow inward from circle 2.
        • +
      +       Another seven-webpage network diff --git a/source/linear-algebra/source/applications/truss.ptx b/source/linear-algebra/source/applications/truss.ptx index defcb3e50..c243dc267 100644 --- a/source/linear-algebra/source/applications/truss.ptx +++ b/source/linear-algebra/source/applications/truss.ptx @@ -11,14 +11,24 @@ In engineering, a truss is a structure designed from several beams of material called struts, assembled to behave as a single object.

      - + + +

      A photograph of a bridge, with the steel girders forming triangles.

      +       + A simple truss
      -
      +
      \drawtruss{} + +

      A diagram representing a simple truss. Points are labeled A,B,C,D,E, with A and B on the same level on top, + and C,D, and E below. The truss is represented by three adjacent triangles: ACD, ADB, and BDE. + Below points C and E are small blue triangles, representing where the bridge is anchored to the ground. + A red arrow points downward from D in the middle of the truss, representing the load put on the truss.

      +       A simple truss
      @@ -38,6 +48,9 @@ with a 10000 N load applied to the node given by D. \drawtruss{} + + The simple truss in is reproduced. + A simple truss
      @@ -69,6 +82,12 @@ some of the struts will be compressed, while others will be tensioned. \drawtruss{\trussCompletion} + + The simple truss in is reproduced, but with additional decorations. + A blue arrow points up and to the right from C, and up and to the left from E. + Edges AB, AC and BE are decorated with red double sided outward arrows indicating tension. + Edges AD and BD are decorated with red double sided inward arrows indicating compression. + Completed truss @@ -89,6 +108,10 @@ For example, at the bottom left node, 3 forces are acting. \drawtruss{\trussCForces} + + The simple truss in is reproduced, but with additional decorations. + At C, red arrows point parallel to the struts towards A and D. A blue arrow points up and to the right from C. + Truss with forces @@ -118,6 +141,9 @@ that must be satisfied for each of the other four nodes of the truss. \drawtruss{} + + The simple truss in is reproduced. + A simple truss @@ -249,6 +275,9 @@ how many scalar variables will be required? \drawtruss{} + + The simple truss in is reproduced. + A simple truss @@ -267,6 +296,10 @@ one variable may be used to represent each. \drawtruss{\trussStrutVariables} + + The simple truss in is reproduced, but now each strut is labeled with a variable. + Struts AB, AC, AD, BD, BE, CD, and DE are labeled x_1, \ldots, x_7 respectively. + Variables for the truss @@ -295,6 +328,10 @@ two variables may be used to represent each. \drawtruss{\trussNormalForces} + + The simple truss in is reproduced, but with additional decorations. + Blue arrows representing normal forces point up and to the right from C, and up and to the left from E. + Truss with normal forces @@ -324,6 +361,9 @@ one for the horizontal component and one for the vertical. \drawtruss{\trussStrutVariables} + + The simple truss in is reproduced, with the variables labeling each strut. + Variables for the truss @@ -359,6 +399,9 @@ then compute each component (approximating \sqrt{3}/2\approx 0.866). \drawtruss{\trussStrutVariables} + + The simple truss in is reproduced, with the variables labeling each strut. + Variables for the truss @@ -463,6 +506,11 @@ The vertical normal forces y_2+z_2 counteract the 10000 load. \drawtruss{\trussCompletion} + + The simple truss in is reproduced, with the red decorations indicationg + tension on struts AC, AB, and BE and compression on struts AD and BD. + The blue normal force vectors pointing up and right from C and up and left from E are also shown. + Completed truss