Chapter 11: Q. 18 (page 901)

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = ⟨ t, α \sin β t, α \cos β t ⟩, t = 0$

Short Answer

Expert verified

Ans: The unit tangent vector to $⟨ t, α \sin β t, α \cos β t ⟩$ at $t = 0$ is $(\frac{1}{\sqrt{1 + α^{2} β^{2}}}, \frac{α β}{\sqrt{1 + α^{2} β^{2}}}, 0)$

Step by step solution

Step 1. Given information:

$r (t) = ⟨ t, α \sin β t, α \cos β t ⟩, t = 0$

Step 2. Simplifying the Unit tangent vectors :

Consider $r (t) = ⟨ t, α \sin β t, α \cos β t ⟩$

First, we compute $r^{'} (t)$

$r^{'} (t) = \frac{d}{d t} ⟨ t, α \sin β t, α \cos β t ⟩ = ⟨ 1, α β \cos β t, - α β \sin β t ⟩ ||r^{'} (t)|| = \sqrt{(1)^{2} + (α β \cos β t)^{2} + (- α β \sin β t)^{2}} = \sqrt{1 + α^{2} β^{2}}$

Step 3. Finding the Unit tangent vectors:

The unit tangent vector to $r (t)$ is

$T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} = \frac{⟨ 1, α β \cos β t, - α β \sin β t ⟩}{\sqrt{1 + α^{2} β^{2}}}$

At $t = 0$ , the unit tangent vector to $r (t)$ is

$T (0) = \frac{⟨ 1, α β \cos 0, - α β \sin 0 ⟩}{\sqrt{1 + α^{2} β^{2}}} = \frac{⟨ 1, α β, 0 ⟩}{\sqrt{1 + α^{2} β^{2}}} = <\frac{1}{\sqrt{1 + α^{2} β^{2}}}, \frac{α β}{\sqrt{1 + α^{2} β^{2}}}, 0>$

Thus the unit tangent vector to $⟨ t, α \sin β t, α \cos β t ⟩$ at $t = 0$ is $(\frac{1}{\sqrt{1 + α^{2} β^{2}}}, \frac{α β}{\sqrt{1 + α^{2} β^{2}}}, 0)$

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Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = ⟨ t, α \sin β t, α \cos β t ⟩, t = 0$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the Unit tangent vectors :

Step 3. Finding the Unit tangent vectors:

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Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.r(t)=⟨t,αsinβt,αcosβt⟩,t=0

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the Unit tangent vectors :

Step 3. Finding the Unit tangent vectors:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = ⟨ t, α \sin β t, α \cos β t ⟩, t = 0$