Chapter 11: Q. 20 (page 901)

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩, t = \frac{π}{3}$

Short Answer

Expert verified

Ans: The principal unit normal vector of $r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩$ at $t = \frac{π}{3}$ is $<\frac{- \sqrt{3}}{2}, \frac{1}{2}>$

Step by step solution

Step 1. Given information:

$r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩, t = \frac{π}{3}$

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:

$r^{'} (t) = ⟨ 6 \cos 2 t, - 6 \sin 2 t ⟩ ||r^{'} (t)|| = \sqrt{(6 \cos 2 t)^{2} + (- 6 \sin 2 t)^{2}} = \sqrt{36 (\cos^{2} 2 t + \sin^{2} 2 t)} = 6$

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

$T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} = \frac{⟨ 6 \cos 2 t, - 6 \sin 2 t ⟩}{6} = ⟨ \cos 2 t, - \sin 2 t ⟩$

Step 3. Finding the principal unit normal vector:

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

$T^{'} (t) = \sqrt{(- 2 \sin 2 t)^{2} + (- 2 \cos 2 t)^{2}} = \sqrt{4 {(\sin^{2} 2 t)}^{2} + (- 2 \cos 2 t)^{2}} = 2$

The principal unit normal vector

$N (t) = \frac{T^{'} (t)}{||T^{'} (t)||} = \frac{⟨ - 2 \sin 2 t, - 2 \cos 2 t ⟩}{2} = ⟨ - 2 \sin 2 t, - \cos 2 t ⟩ At t = \frac{π}{3}, N (t) = N (\frac{π}{3}) = <- \sin \frac{2 π}{3}, - \cos \frac{2 π}{3}> = <\frac{- \sqrt{3}}{2}, \frac{1}{2}>$

Thus the principle unit normal vector of $r (t)$ at $t = \frac{π}{3}$ is

$<\frac{- \sqrt{3}}{2}, \frac{1}{2}>$

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Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩, t = \frac{π}{3}$

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)=⟨3sin2t,3cos2t⟩,t=π3

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

Pure Maths

Applied Mathematics

Calculus

Logic and Functions

Probability and Statistics

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.
$r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩, t = \frac{π}{3}$