Arc Length with Vector Functions — Question 10

Question 10 – Solution

Strategy. Express displacement as the integral of the velocity vector and apply a dot-product estimate in the displacement direction.

Step 1: Zero-displacement case If $\mathbf r(b)=\mathbf r(a)$, the left side is zero and the inequality follows because the integral of speed is nonnegative.

Now suppose $\mathbf d=\mathbf r(b)-\mathbf r(a)\ne\mathbf 0$ and define the unit vector $\mathbf u=\mathbf d/\|\mathbf d\|$.

Step 2: Use the Fundamental Theorem $$\mathbf d=\int_a^b\mathbf r'(t)\,dt.$$ Dot both sides with $\mathbf u$: $$\begin{align*} \|\mathbf d\| &=\mathbf d\cdot\mathbf u\\ &=\int_a^b\mathbf r'(t)\cdot\mathbf u\,dt\\ &\le\int_a^b|\mathbf r'(t)\cdot\mathbf u|\,dt\\ &\le\int_a^b\|\mathbf r'(t)\|\,\|\mathbf u\|\,dt\\ &=\int_a^b\|\mathbf r'(t)\|\,dt. \end{align*}$$ Therefore $$\boxed{\|\mathbf r(b)-\mathbf r(a)\|\le \int_a^b\|\mathbf r'(t)\|\,dt}.$$

Step 3: Geometry and equality The straight chord is the shortest connection between the endpoints; following a curved or reversing path cannot be shorter. If $\mathbf r'=\lambda\mathbf u$ with $\lambda\ge 0$, then $$\mathbf r(b)-\mathbf r(a)=\mathbf u\int_a^b\lambda(t)\,dt,$$ so its norm is $\int_a^b\lambda(t)dt$. Meanwhile $\|\mathbf r'\|=\lambda$, giving equality. The condition describes travel in one fixed direction without backtracking.