\h1{Geometry}
\h2{Lorem \{x^2} Ipsum}
\h2{Definition of a Line}
\img[width=500px center src=../static/drawings/matrix/vector-equation-of-a-line.png]
\note[boxed] {
\h3{Symmetric Equation of a Line}
Given
\equation {
t &= \frac{x - x_1}{x_2-x_1} = \frac{x - x_1}{\Delta_x}\\
t &= \frac{y - y_1}{y_2-y_1} = \frac{y - y_1}{\Delta_y}\\
t &= \frac{z - z_1}{z_2-z_1} = \frac{z - z_1}{\Delta_z}
}
Therefore
\equation {
\frac{x - x_1}{Delta_x}
&= \frac{y - y_1}{\Delta_y}
= \frac{z - z_1}{\Delta_z}\\
\frac{x - x_1}{x_2-x_1}
&= \frac{y - y_1}{y_2-y_1}
= \frac{z - z_1}{z_2-z_1}
}
\hr
\h4{Rationale}
We rewrite \{r = r_0 + a = r_0 + t v} in terms of \{t}.
That is
\equation{
x &= x_1 + t(x_2-x_1) = x_1 + t\;Delta_x\\
t\;Delta_x &= x - x_1 = t(x_2-x_1)\\
t &= \frac{x - x_1}{x_2-x_1} = \frac{x - x_1}{Delta_x} \\\\
y &= y_1 + t(y_2-y_1) = y_1 + t\;\Delta_y\\
t\;\Delta_y &= y - y_1 = t(y_2-y_1)\\
t &= \frac{y - y_1}{y_2-y_1} = \frac{y - y_1}{\Delta_y} \\\\
z &= z_1 + t(z_2-z_1) = z_1 + t\;\Delta_z\\
t\;\Delta_z &= z - z_1 = t(z_2-z_1) \\
t &= \frac{z - z_1}{z_2-z_1} = \frac{z - z_1}{\Delta_z}
}
}
\!where {
{\Delta_x} => {\colorA{\Delta_x}}
{\Delta_y} => {\colorA{\Delta_y}}
{\Delta_z} => {\colorA{\Delta_z}}
{x_1} => {\colorB{x_1}}
{y_1} => {\colorB{y_1}}
{z_1} => {\colorB{z_1}}
}
\note {
\h1{The Acceleration Vector \{\vec{v}} }
\equation {
\vec{a} =
\lim_{t\to 0} \frac{\vec{v}(t + \Delta t)-\vec{v}}{\Delta t} =
\frac
{\mathrm{d}\vec{v}}
{\mathrm{d}t}
}
}