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Added equations to tuts 0, 8, 9, and 10.
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# File Documents/Basics/ColumnVector.mathml

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`+[/itex]`

# File Documents/Basics/Tutorial 00.xml

`         <para>When writing vectors in text, they are written with parenthesis. So a 3D vector could`
`             be (0, 2, 4); the X component is 0, the Y component is 2, and the Z component is 4. When`
`             writing them as part of an equation, they are written as follows:</para>`
`-        <!--TODO: Show a column vector.-->`
`-        <para>In math equations, vector variables are either bold or written with an arrow over`
`-            them.</para>`
`+        <informalequation>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="ColumnVector.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </informalequation>`
`+        <para>In math equations, vector variables are either in boldface or written with an arrow`
`+            over them.</para>`
`         <para>When drawing vectors graphically, one makes a distinction between position vectors and`
`             direction vectors. However, numerically there is <emphasis>no</emphasis> difference`
`             between the two. The only difference is in how you use them, not how you represent them`
`             <!--TODO: Show A+B again, with B moved to the tip of A.-->`
`             <para>Numerically, the sum of two vectors is just the sum of the corresponding`
`                 components:</para>`
`-            <!--TODO: Show vector addition in equation form.-->`
`+        <equation>`
`+            <title>Vector Addition with Numbers</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorAdditionNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`             <para>Any operation where you perform an operation on each component of a vector is`
`                 called a <glossterm>component-wise operation</glossterm>. Vector addition is`
`                 component-wise. Any component-wise operation on two vectors requires that the two`
`         </formalpara>`
`         <!--TODO: Show vectors A and -A.-->`
`             <para>Numerically, this means negating each component of the vector.</para>`
`-            <!--TODO: Show negation in equation form.-->`
`-            <para>Just as with regular math, vector subtraction is the same as addition with the`
`-                negation of the second vector.</para>`
`+        <equation>`
`+            <title>Vector Negation</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorNegationNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`+            <para>Just as with scalar math, vector subtraction is the same as addition with the`
`+            negation of the second vector.</para>`
`             <!--TODO: Geometrically show vector subtraction.-->`
`         <formalpara>`
`             <title>Vector Multiplication</title>`
`-            <para>Vector multiplication is one of the few vector operations that has no geometric`
`-                equivalent. Multiplying two vectors numerically is simply component-wise`
`-                multiplication, much like vector addition.</para>`
`+            <para>Vector multiplication is one of the few vector operations that has no real`
`+                geometric equivalent. To multiply a direction by another, or multiplying a position`
`+                by another position, doesn't really make sense. That doesn't mean that the numerical`
`+                equivalent isn't useful, though.</para>`
`         </formalpara>`
`-        <!--TODO: Numerically show vector multiplication.-->`
`+        <para>Multiplying two vectors numerically is simply component-wise multiplication, much like`
`+            vector addition.</para>`
`+        <equation>`
`+            <title>Vector Multiplication</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorMultiplicationNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`         <formalpara>`
`             <title>Vector/Scalar Operations</title>`
`             <para>Vectors can be operated on by scalar values. Recall that scalars are just single`
`         <!--TODO: Graph of a vector, that vector times 3, and that vector times 0.5.-->`
`         <para>Numerically, this is a component-wise multiplication, where each component of the`
`             vector is multiplied with each component of the scalar.</para>`
`-        <!--TODO: Equation of vector-scalar multiplication.-->`
`-        <para>Scalars can also be added to scalars. This, like vector-to-vector multiplication, has`
`+        <equation>`
`+            <title>Vector-Scalar Multiplication</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorScalarMultNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`+        <para>Scalars can also be added to vectors. This, like vector-to-vector multiplication, has`
`             no geometric representation. It is a component-wise addition of the scalar with each`
`             component of the vector.</para>`
`-        <!--TODO: Equation of vector-scalar addition.-->`
`+        <equation>`
`+            <title>Vector-Scalar Addition</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorScalarAddNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`         <formalpara>`
`             <title>Vector Algebra</title>`
`             <para>It is useful to know a bit about the relationships between these kinds of vector`
`                 operations.</para>`
`         </formalpara>`
`-        <para>Vector addition and multiplication follow the same rules for scalar addition and`
`-            multiplication. They are commutative, associative, and distributive.</para>`
`-        <!--TODO: Equation for commutative, associative, and distributive vector math.-->`
`-        <para>Scalar operations have similar properties.</para>`
`+        <para>Vector addition and multiplication follow many of the same rules for scalar addition`
`+            and multiplication. They are commutative, associative, and distributive.</para>`
`+        <equation>`
`+            <title>Vector Algebra</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorMathProperties.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`+        <para>Vector/scalar operations have similar properties.</para>`
`         <formalpara>`
`             <title>Length</title>`
`             <para>Vectors have a length. The length of a vector is the distance from the starting`
`         </formalpara>`
`         <!--TODO: Show a graph of a vector and its length.-->`
`         <para>Numerically, computing the distance requires this equation:</para>`
`-        <!--TODO: Equation for distance computation.-->`
`+        <equation>`
`+            <title>Vector Length</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorLengthNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`         <formalpara>`
`             <title>Unit Vectors and Normalization</title>`
`             <para>A vector that has a length of exactly one is called a <glossterm>unit`
`         <para>A vector can be converted into a unit vector by <glossterm>normalizing</glossterm> it.`
`             This is done by dividing the vector by its length. Or rather, multiplication by the`
`             reciprocal of the length.</para>`
`-        <!--TODO: Equation for normalization of a vector.-->`
`+        <equation>`
`+            <title>Vector Normalization</title>`
`+            <mediaobject>`
`+                <imageobject>`
`+                    <imagedata format="SVG" fileref="VectorNormalizationNum.svg"/>`
`+                </imageobject>`
`+            </mediaobject>`
`+        </equation>`
`         <para>This is not all of the vector math that we will use in these tutorials. New vector`
`             math operations will be introduced and explained as needed when they are first used. And`
`             unlike the math operations introduced here, most of them are not component-wise`

# File Documents/Basics/Tutorial 01.xml

`                 position, this shader simply copies it directly into the output.</para>`
`             <formalpara>`
`                 <title>Vertex Attributes</title>`
`+                <para>Shaders have inputs and outputs. Think of these like function parameters and`
`+                    function return values. If the shader is a function, then it is called with`
`+                    input values, and it is expected to return a number of output values.</para>`
`             </formalpara>`
`-            <para>Shaders have inputs and outputs. Think of these like function parameters and`
`-                function return values. If the shader is a function, then it is called with input`
`-                values, and it is expected to return a number of output values.</para>`
`             <para>Inputs to and outputs from a shader stage come from somewhere and go to somewhere.`
`                 Thus, the input <varname>position</varname> in the vertex shader must be filled in`
`                 with data somewhere. So where does that data come from? Inputs to a vertex shader`

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# File Documents/Basics/VectorLengthNum.mathml

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# File Documents/Basics/VectorMathProperties.mathml

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`+  <mrow>`
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`+      <mtd>`
`+        <mtext>Commutative</mtext>`
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`+        <mover><mi>a</mi><mo>&#x21C0;</mo></mover>`
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`+        <mtext>Associative</mtext>`
`+      </mtd>`
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`+[/itex]`

# File Documents/Basics/VectorMultiplicationNum.mathml

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`+[/itex]`

# File Documents/Basics/VectorNegationNum.mathml

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# File Documents/Illumination/HalfLightAtten.mathml

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