Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? I'm not sure where these steps are coming from.
Example:
Show that in $\mathbb{R}^n$, the length of any parametric curve connecting a and b is at least as long as the length of the straight line connecting the points with position vectors a and b.
Solution:
For a parametric curve with $I = [a,b]$ and x(a) = a and x(b) = b and for a unit vector u we have the inequality $$({\bf{b}-{\bf{a}}})\cdot {\bf{u}} = \int_a^b \dot {{\bf{x}}} \cdot {\bf{u}} {\it{dt}} ≤\int_ b ^a |\dot{\bf{x}}|dt \quad (*)$$
as $\dot{\bf{x}}·{\bf{u} = |\dot{x}||u|}\cosθ ≤| {\bf{\dot{x}}}||{\bf{u}}| = |\ {\bf{\dot{x}}}|$, and by taking $${\bf{u}} =\frac{ {\bf{b−a} }}{{\bf{|b−a|}}} \quad @$$ we obtain that $$|{\bf{b−a}}|≤\int_{b}^ a |\dot{\bf{ x}}| dt = L $$so that the length of the curve is at least |b−a|.
The dot product is taken because you want to estimate the length $|{\bf b} - {\bf a}|$.
Now:
$$|{\bf b} - {\bf a}|^2 = ({\bf b}-{\bf a}) \cdot ({\bf b} - {\bf a})$$
Thus:
$$|{\bf b} - {\bf a}| = \frac{({\bf b}-{\bf a}) \cdot ({\bf b} - {\bf a}) }{ |{\bf b} - {\bf a}|} = ({\bf b}-{\bf a}) \cdot {\bf u}$$
Note that the unit vector $\bf{u}$ is constant. If you spell out the inner product, you'll get
$$ \int_a^b {\dot{\bf{x}}} {\cdot} {\bf{u}} \, dt = \int_a^b \sum_{i=1}^n \frac {dx_i(t)} {dt} u_i \, dt = \sum_{i=1}^n u_i \int_a^b \frac {dx_i(t)}{dt} dt $$
The integral on the right amounts to the total difference $x_i(b) - x_i(a)$. Thus,
$$ \sum_{i=1}^n u_i \int_a^b \frac {dx_i(t)}{dt} dt = \sum_{i=1}^n u_i \, (b_i - a_i) = {\bf u} \cdot ({\bf b} - {\bf a}) $$