Chapter 11: Q. 21 (page 901)

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = ⟨ α \sin β t, α \cos β t ⟩$ , where $α$ and $β$ are positive, $t = 0$

Short Answer

Expert verified

Ans: The principal unit normal vector of $r (t) = ⟨ α \sin β t, α \cos β t ⟩$ at $t = 0$ is $⟨ 0, - 1 ⟩$

Step by step solution

Step 1. Given information:

$r (t) = ⟨ α \sin β t, α \cos β t ⟩$ , where $α$ and $β$ are positive, $t = 0$

Step 2. Simplifying the principal unit normal vector:

The principal unit normal vector of $r (t)$ denoted by $N (t)$ is given by

$N (t) = \frac{T^{'} (t)}{||T^{'} (t)||}$ where $T (t) = \frac{r^{'} (t)}{||r^{'} (t)||}$

$T (t)$ should be calculated first and then $N (t)$ is to be evaluated.

We start by comparing $r^{'} (t)$ first:

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

The unit tangent vector

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

To find $N (t)$ we divide $T^{'} (t)$ by its magnitude.

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

Step 3. Finding the principal unit normal vector:

The principal unit normal vector

$N (t) = \frac{T^{'} (t)}{||T^{'} (t)||} = \frac{⟨ - β \sin β t, - β \cos β t ⟩}{β} = ⟨ - \sin β t, - \cos β t ⟩$

At $t = 0, N (t) = N (0) = ⟨ - \sin 0, - \cos 0 ⟩ = ⟨ 0, - 1 ⟩$

Thus the principle unit normal vector of $r (t)$ at $t = 0$ is $⟨ 0, - 1 ⟩$

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Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = ⟨ α \sin β t, α \cos β t ⟩$ , where $α$ and $β$ are positive, $t = 0$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the principal unit normal vector:

Step 3. Finding the principal unit normal vector:

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Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.r(t)=⟨αsinβt,αcosβt⟩, where α and β are positive,t=0

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the principal unit normal vector:

Step 3. Finding the principal unit normal vector:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Calculus

Study anywhere. Anytime. Across all devices.

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = ⟨ α \sin β t, α \cos β t ⟩$ , where $α$ and $β$ are positive, $t = 0$