Question: "Unit tangent vectors Find the unit tangent vector..."

High School
Calculus and Analysis
Calculus 2
Parametric equations

rompentskj

Answered question

2021-12-01

Unit tangent vectors Find the unit tangent vector for the following parameterized curves.

r (t) = ⟨ 8, \cos 2 t, 2 \sin 2 t ⟩

, for

0 \leq t \leq 2 π

Answer & Explanation

Twereen

Beginner2021-12-02Added 13 answers

So,
$r (t) = ⟨ 8, \cos 2 t, 2 \sin 2 t ⟩$ , for $0 \leq t \leq 2 π$
$\frac{d (\sin (a x))}{d x} = a \cos (a x)$
$\frac{d (\cos (a x))}{d x} = - a \sin (a x)$
$r ‘ (t) = ⟨ 0, - 2 \sin 2 t, 2 \cos 2 t ⟩$
$| r ‘ (t) | = \sqrt{0^{2} + {(- 2 \sin 2 t)}^{2} + {(2 \cos 2 t)}^{2}}$
$= \sqrt{4 \sin^{22} t + 4 \cos^{22} t}$
$= \sqrt{4 (\sin^{22} t + \cos^{22} t)}$
$= 2$
$\vec{T} (t) = \frac{r ‘ (t)}{| r ‘ (t) |}$
$= \frac{⟨ 0, - 2 \sin 2 t, 2 \cos 2 t ⟩}{2}$
$= \frac{1}{3} (0 i - 2 \sin 2 t j + 2 \cos 2 t k)$
$= \frac{1}{3} (- 2 \sin 2 t j + 2 \cos 2 t k)$

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