Chapter 11: Q. 19 (page 901)

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = <t, t^{3}>, t = 2$

Short Answer

Expert verified

Ans: Thus the principal unit normal vector of $r (t) = <t, t^{3}>$ at $t = 2$ is $(\frac{- 12 \sqrt{145}}{145}, \frac{\sqrt{145}}{145})$

Step by step solution

Step 1. Given information:

$r (t) = <t, t^{3}>, t = 2$

Step 2. Simplifying the principal unit normal vector:

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

$r^{'} (t) = <1, 3 t^{2}> ||r^{'} (t)|| = \sqrt{1 + {(3 t^{2})}^{2}} = \sqrt{1 + 9 t^{4}}$

The unit tangent vector of $r (t)$

$T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} = \frac{<1, 3 t^{2}>}{\sqrt{1 + 9 t^{4}}} T^{'} (t) = \sqrt{\frac{- 18 t^{3}}{{(1 + 9 t^{4})}^{\frac{3}{2}}}, \frac{6 t}{{(1 + 9 t^{4})}^{\frac{3}{2}}})} use the Quotient rule ||T^{'} (t)|| = \sqrt{{(\frac{- 18 t^{3}}{{(1 + 9 t^{4})}^{\frac{3}{2}}})}^{2} + {(\frac{6 t}{{(1 + 9 t^{4})}^{\frac{3}{2}}})}^{2}} = \sqrt{\frac{324 t^{6} + 36 t^{2}}{{(1 + 9 t^{4})}^{3}}} \frac{6 t}{1 + 9 t^{4}}$

Step 3. Finding the principal unit normal vector:

The principal unit normal vector

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

$= \frac{1 + 9 t^{4}}{6 t} <\frac{- 18 t^{3}}{{(1 + 9 t^{4})}^{\frac{3}{2}}}, \frac{6 t}{{(1 + 9 t^{4})}^{\frac{3}{2}}}>$

$= <\frac{- 3 t^{2}}{{(1 + 9 t^{4})}^{\frac{1}{2}}}, \frac{1}{{(1 + 9 t^{4})}^{\frac{1}{2}}}>$

$N (t) = N (2) = <\frac{- 3 (2)^{2}}{{(1 + 9 (2)^{4})}^{\frac{1}{2}}}, \frac{1}{{(1 + 9 (2)^{4})}^{\frac{1}{2}}}>$

$Att = 2$

$= <\frac{- 12}{\sqrt{145}}, \frac{1}{\sqrt{145}}>$

$= <\frac{- 12 \sqrt{145}}{145}, \frac{\sqrt{145}}{145}>$

Thus the principal unit normal vector of $r (t)$ at $t = 2$ is $(\frac{- 12 \sqrt{145}}{145}, \frac{\sqrt{145}}{145})$

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Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = <t, t^{3}>, t = 2$

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)=t,t3,t=2

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

Applied Mathematics

Decision Maths

Mechanics Maths

Geometry

Pure Maths

Discrete Mathematics

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.
$r (t) = <t, t^{3}>, t = 2$