Find an expression for a unit vector normal to the surfacex = 10 cos (0) sin ($) , y = 5 sin (0) sin (4) , z = cos (4)for 0 in [0, 27] and o in [0, x].(Enter your solution in the vector form (*,*,*). Use symbolic notation and fractions where needed.)n =

Answered: Find an expression for a unit vector…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 45E

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Question

Find an expression for a unit vector normal to the surface
x = 10 cos (0) sin ($) , y = 5 sin (0) sin (4) , z = cos (4)
for 0 in [0, 27] and o in [0, x].
(Enter your solution in the vector form (*,*,*). Use symbolic notation and fractions where needed.)
n =

Expert Solution

Step 1

Given that $x = 10 \cos (θ) \sin (ϕ)$ , $y = 5 \sin (θ) \sin (ϕ)$ and $z = \cos (ϕ)$ .

We have to find a unit vector normal to the surface.

Let $r (θ, ϕ) = 〈10 \cos (θ) \sin (ϕ), 5 \sin (θ) \sin (ϕ), \cos (ϕ)〉$

Now, $r_{θ} = 〈- 10 \sin (θ) \sin (ϕ), 5 \cos (θ) \sin (ϕ), 0〉$

And $r_{ϕ} = 〈10 \cos (θ) \cos (ϕ), 5 \sin (θ) \cos (ϕ), - \sin (ϕ)〉$ .

$\begin{matrix} r_{θ} \times r_{ϕ} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 10 \sin (θ) \sin (ϕ) & 5 \cos (θ) \sin (ϕ) & 0 \\ 10 \cos (θ) \cos (ϕ) & 5 \sin (θ) \cos (ϕ) & - \sin (ϕ) \end{matrix}| \\ = \hat{i} (- 5 \cos (θ) \sin^{2} (ϕ) - 0) - \hat{j} (10 \sin (θ) \sin^{2} (ϕ) - 0) + \hat{k} (- 50 \sin^{2} (θ) \sin (ϕ) \cos (ϕ) - 50 \cos^{2} (θ) \sin (ϕ) \cos (ϕ)) \\ = \hat{i} (- 5 \cos (θ) \sin^{2} (ϕ)) - \hat{j} (10 \sin (θ) \sin^{2} (ϕ)) + \hat{k} (- 50 \sin (ϕ) \cos (ϕ)) \\ = 〈- 5 \cos (θ) \sin^{2} (ϕ), - 10 \sin (θ) \sin^{2} (ϕ), - 50 \sin (ϕ) \cos (ϕ)〉 \end{matrix}$

Hence, $r_{θ} \times r_{ϕ} = 〈- 5 \cos (θ) \sin^{2} (ϕ), - 10 \sin (θ) \sin^{2} (ϕ), - 50 \sin (ϕ) \cos (ϕ)〉$ .

Now,

$\begin{matrix} ||r_{θ} \times r_{ϕ}|| = \sqrt{{(- 5 \cos (θ) \sin^{2} (ϕ))}^{2} + {(- 10 \sin (θ) \sin^{2} (ϕ))}^{2} + {(- 50 \sin (ϕ) \cos (ϕ))}^{2}} \\ = \sqrt{25 \cos^{2} (θ) \sin^{4} (ϕ) + 100 \sin^{2} (θ) \sin^{4} (ϕ) + 2500 \sin^{2} (ϕ) \cos^{2} (ϕ)} \\ = \sqrt{25 \sin^{2} (ϕ) (\cos^{2} (θ) \sin^{2} (ϕ) + 4 \sin^{2} (θ) \sin^{2} (ϕ) + 100 \cos^{2} (ϕ))} \\ = 5 \sin (ϕ) \sqrt{(\cos^{2} (θ) \sin^{2} (ϕ) + 4 \sin^{2} (θ) \sin^{2} (ϕ) + 100 \cos^{2} (ϕ))} \end{matrix}$

Hence, $||r_{θ} \times r_{ϕ}|| = 5 \sin (ϕ) \sqrt{(\cos^{2} (θ) \sin^{2} (ϕ) + 4 \sin^{2} (θ) \sin^{2} (ϕ) + 100 \cos^{2} (ϕ))}$ .

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Find an expression for a unit vector normal to the surface x = 10 cos (0) sin ($), y = 5 sin (0) sin (4), z = cos (4) for 0 in [0, 2x] and ø in [0, x]. (Enter your solution in the vector form (*,*,*). Use symbolic notation and fractions where needed.) n =

Find an expression for a unit vector normal to the surface x = 10 cos (0) sin ($), y = 5 sin (0) sin (4), z = cos (4) for 0 in [0, 2x] and ø in [0, x]. (Enter your solution in the vector form (,,*). Use symbolic notation and fractions where needed.) n =