Chapter 11: Q. 23 (page 901)

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

Short Answer

Expert verified

The principal unit normal vector to $r (t)$ is,

$N (t) = \frac{1}{\sqrt{(4 π^{2} + 5) (4 π^{4} + 17 π^{2} + 20)}} <- 2 π, - 2 (5 + 2 π^{2}), π (9 + 4 π^{2})>$ at $t = π$ .

Step by step solution

Step 1. Given Information

We are given an function,

$r (t) = (t, 2 t \sin t, 2 t \cos t)$

And,

role="math" $t = π$

Step 2. Finding principal unit normal vector

Principal unit normal vector:

If $r (t)$ is a twice differentiable vector function, then the principal unit vector to $r (t)$ is

localid="1649652059482" $N (t) = \frac{T^{'} (t)}{|T^{'} (t)|}$

$r (t) = ⟨ t, 2 t \sin t, 2 t \cos t ⟩ r^{'} (t) = ⟨ 1, 2 (t \cos t + \sin t), 2 (- t \sin t + \cos t) ⟩ |r^{'} (t)| = \sqrt{1 + 4 (t \cos t + \sin t)^{2} + 4 (- t \sin t + \cos t)^{2}} = \sqrt{1 + 4 (t^{2} + 1)} = \sqrt{4 t^{2} + 5}$

Step 3. Finding principal unit normal vector

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

role="math" localid="1649653047945" $T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} T (t) = \frac{1}{\sqrt{4 t^{2} + 5}} ⟨ 1, 2 (t \cos t + \sin t), 2 (- t \sin t + \cos t) ⟩ T (t) = <\frac{1}{\sqrt{4 t^{2} + 5}}, \frac{2 (t \cos t + \sin t)}{\sqrt{4 t^{2} + 5}}, \frac{2 (- t \sin t + \cos t)}{\sqrt{4 t^{2} + 5}}>$

Using the Quotient rule for derivatives,

$T^{'} (t) = <\begin{array}{l} \frac{- 4 t}{{(4 t^{2} + 5)}^{\frac{3}{2}}}, \frac{2 (4 t^{2} + 5) (- t \sin t + 2 \cos t) - 8 t (t \cos t + \sin t)}{{(4 t^{2} + 5)}^{\frac{3}{2}}} \\ , \frac{2 (4 t^{2} + 5) (- t \cot - 2 \sin t) - 8 t (- t \sin t + \cos t)}{{(4 t^{2} + 5)}^{\frac{3}{2}}} \end{array}> ||T^{'} (t)|| = \sqrt{\frac{16 t^{2} + 4 {(4 t^{2} + 5)}^{2} (t^{2} + 4) + 64 t^{2} (t^{2} + 1) - 32 t^{2} (4 t^{2} + 5)}{{(4 t^{2} + 5)}^{3}}} ||T^{'} (t)|| = \sqrt{\frac{64 t^{6} + 352 t^{4} + 660 t^{2} + 400}{{(4 t^{2} + 5)}^{3}}} ||T^{'} (t)|| = \sqrt{\frac{4 (16 t^{6} + 88 t^{4} + 165 t^{2} + 100)}{{(4 t^{2} + 5)}^{3}}} ||T^{'} (t)|| = \frac{2 \sqrt{16 t^{6} + 88 t^{4} + 165 t^{2} + 100}}{{(4 t^{2} + 5)}^{\frac{3}{2}}} ||T^{'} (t)|| = \frac{2 \sqrt{(4 t^{2} + 5) (4 t^{4} + 17 t^{2} + 20)}}{{(4 t^{2} + 5)}^{\frac{3}{2}}} N (t) = \frac{T^{'} (t)}{||T^{'} (t)||}$

Step 4. Finding principal unit normal vector

We need to evaluate N(t) at $t = π$ ,

$T^{'} (t) = <\frac{- 4 π}{{(4 π^{2} + 5)}^{\frac{3}{2}}}, \frac{2 (4 π^{2} + 5) (- 2) - 8 π (- π)}{{(4 π^{2} + 5)}^{\frac{3}{2}}}, \frac{2 (4 π^{2} + 5) (π) - 8 π (- 1)}{{(4 π^{2} + 5)}^{\frac{3}{2}}}> T^{'} (t) = <\frac{- 4 π}{{(4 π^{2} + 5)}^{\frac{3}{2}}}, \frac{- 8 π^{2} - 20}{{(4 π^{2} + 5)}^{\frac{3}{2}}}, \frac{8 π^{3} + 18 π}{{(4 π^{2} + 5)}^{\frac{3}{2}}}> |T^{'} (t)| = \frac{2 \sqrt{(4 π^{2} + 5) (4 π^{4} + 17 π^{2} + 20)}}{{(4 π^{2} + 5)}^{\frac{3}{2}}}$

So,

role="math" localid="1649654290745" $N (t) = \frac{T^{'} (t)}{|T^{'} (t)|} = <\frac{- 2 π}{\sqrt{(4 π^{2} + 5) (4 π^{4} + 17 π^{2} + 20)}}, \frac{- 2 (5 + 2 π^{2})}{\sqrt{(4 π^{2} + 5) (4 π^{4} + 17 π^{2} + 20)}}, \frac{π (9 + 4 π^{2})}{\sqrt{(4 π^{2} + 5) (4 π^{4} + 17 π^{2} + 20)}}>$

Thus, the principal unit normal vector is,

$N (t) = \frac{1}{\sqrt{(4 π^{2} + 5) (4 π^{4} + 17 π^{2} + 20)}} <- 2 π, - 2 (5 + 2 π^{2}), π (9 + 4 π^{2})>$

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding principal unit normal vector

Step 3. Finding principal unit normal vector

Step 4. Finding 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,2tsint,2tcost),t=π

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding principal unit normal vector

Step 3. Finding principal unit normal vector

Step 4. Finding principal unit normal vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

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.
$r (t) = (t, 2 t \sin t, 2 t \cos t), t = π$