Perpendicular tangent vector
by Hidenori
Proposition
Let $c:[0,1] \rightarrow \mathbb{R}^n$ be a curve such that $\abs{c(t)} = 1$ for all $t$. Show that $c(t)_{c(t)}$ and the tangent vector to $c$ at $t$ are perpendicular.
Solution
\[\begin{align*} \ev{(c^1(t), \cdots, c^n(t))_{c(t)}, ((c^1)'(t), \cdots, (c^n)'(t))_{c(t)}} &= \ev{(c^1(t), \cdots, c^n(t)), ((c^1)'(t), \cdots, (c^n)'(t))} \\ &= c^1(t)(c^1)'(t) + \cdots + c^n(t)(c^n)'(t) \\ &= D(c^1)^2(t) + \cdots + D(c^n)^2(t) \\ &= D((c^1)^2 + \cdots + (c^n)^2)(t) \\ &= 0. \end{align*}\]Subscribe via RSS