Short Answer

Let C be the graph of a vector-valued function r. The plane determined by the vectors T(t₀) and B(t₀) and containing the point r(t₀) is called the rectifying plane for C at r(t₀). Find the equation of the rectifying plane to the helix determined by $r (t) = (\cos (t), \sin (t), t)$ when t = π.

$x + 1 = 0$

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TABLE OF CONTENTS

Step1. Given Information

$Consider r (t) = ⟨ \cos t, \sin t, t ⟩ at t = \frac{π}{2} The objective is to find the equations of the rectifying plane to r (t) at t = π . For this, T (t), N (t) and B (t) should be calculated first. Consider \begin{array}{l} r (t) = ⟨ \cos t, \sin t, t ⟩ \\ r^{'} (t) = ⟨ - \sin t, \cos t, 1 ⟩ \\ \begin{aligned} ∥r^{'} (t)∥ = \sqrt{\sin^{2} t + \cos^{2} t + 1} \\ = \sqrt{2} \end{aligned} \\ T (t) = \frac{r^{'} (t)}{∥r^{'} (t)∥} \\ = \frac{⟨ - \sin t, \cos t, 1 ⟩}{\sqrt{2}} \\ At t = π, T (t) = T (π) = \frac{⟨ - \sin π, \cos π, 1 ⟩}{\sqrt{2}} \\ = \frac{⟨ 0, - 1, 1 ⟩}{\sqrt{2}} \\ = <0, \frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \\ Thus the unit tangent vector to r (t) at t = π is \\ \begin{array}{l} T (π) = <0, \frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \\ T^{'} (t) = \frac{1}{\sqrt{2}} ⟨ - \cos t, - \sin t, 0 ⟩ \\ ∥T^{'} (t)∥ = \sqrt{\frac{1}{2} \cos^{2} t + \frac{1}{2} \sin^{2} t} \\ = \frac{1}{\sqrt{2}} \\ N (t) = \frac{T^{'} (t)}{∥T^{'} (t)∥} = \frac{\frac{1}{\sqrt{2}} ⟨ - \cos t, - \sin t, 0 ⟩}{\frac{- 1}{\sqrt{2}}} \\ = ⟨ - \cos t, - \sin t, 0 ⟩ \\ At t = π, N (t) = N (π) = ⟨ - \cos π, - \sin π, 0 ⟩ \\ = ⟨ 1, 0, 0 ⟩ \\ Thus the principal unit normal vector to r (t) at t = π is ⟨ 1, 0, 0 ⟩ \\ B (π) = T (π) \times N (π) \\ \begin{array}{l} = |\begin{array}{ccc} i & j & k \\ 0 & \frac{- \sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ 1 & 0 & 0 \end{array}| \\ = i (0) - j (\frac{- \sqrt{2}}{2}) + k (\frac{\sqrt{2}}{2}) \\ = <0, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \end{array} \end{array} \end{array}$ uncaught exception: Invalid chunkin file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('3012ea183b02056...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage(' $" width="0" height="0" role="math" style="max-width: none;" localid="1649571995194"> Consider r (t) = ⟨ \cos t, \sin t, t ⟩ at t = \frac{π}{2} The objective is to find the equations of the rectifying plane to r (t) at t = π . For this, T (t), N (t) and B (t) should be calculated first. r (t) = <e^{t}, e^{- t}, \sqrt{2} t) at t = 0 Consider \begin{array}{l} r (t) = ⟨ \cos t, \sin t, t ⟩ \\ r^{'} (t) = ⟨ - \sin t, \cos t, 1 ⟩ \\ \begin{aligned} ∥r^{'} (t)∥ = \sqrt{\sin^{2} t + \cos^{2} t + 1} \\ = \sqrt{2} \end{aligned} \\ T (t) = \frac{r^{'} (t)}{∥r^{'} (t)∥} \\ = \frac{⟨ - \sin t, \cos t, 1 ⟩}{\sqrt{2}} \\ At t = π, T (t) = T (π) = \frac{⟨ - \sin π, \cos π, 1 ⟩}{\sqrt{2}} \\ = \frac{⟨ 0, - 1, 1 ⟩}{\sqrt{2}} \\ = <0, \frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \\ Thus the unit tangent vector to r (t) at t = π is \\ \begin{array}{l} T (π) = <0, \frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \\ T^{'} (t) = \frac{1}{\sqrt{2}} ⟨ - \cos t, - \sin t, 0 ⟩ \\ ∥T^{'} (t)∥ = \sqrt{\frac{1}{2} \cos^{2} t + \frac{1}{2} \sin^{2} t} \\ = \frac{1}{\sqrt{2}} \\ N (t) = \frac{T^{'} (t)}{∥T^{'} (t)∥} = \frac{\frac{1}{\sqrt{2}} ⟨ - \cos t, - \sin t, 0 ⟩}{\frac{- 1}{\sqrt{2}}} \\ = ⟨ - \cos t, - \sin t, 0 ⟩ \\ At t = π, N (t) = N (π) = ⟨ - \cos π, - \sin π, 0 ⟩ \\ = ⟨ 1, 0, 0 ⟩ \\ Thus the principal unit normal vector to r (t) at t = π is ⟨ 1, 0, 0 ⟩ \\ B (π) = T (π) \times N (π) \\ \begin{array}{l} = |\begin{array}{ccc} i & j & k \\ 0 & \frac{- \sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ 1 & 0 & 0 \end{array}| \\ = i (0) - j (\frac{- \sqrt{2}}{2}) + k (\frac{\sqrt{2}}{2}) \\ = <0, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \end{array} \end{array} \end{array} uncaught exception:$ Invalid chunkin file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('c83cb95b9d1d76b...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage(' $" width="0" height="0" role="math" style="max-width: none;" localid="1649572366709"> Consider r (t) = ⟨ \cos t, \sin t, t ⟩ at t = \frac{π}{2} The objective is to find the equations of the rectifying plane to r (t) at t = π . For this, T (t), N (t) and B (t) should be calculated first. r (t) = <e^{t}, e^{- t}, \sqrt{2} t) at t = 0 Consider \begin{array}{l} r (t) = ⟨ \cos t, \sin t, t ⟩ \\ r^{'} (t) = ⟨ - \sin t, \cos t, 1 ⟩ \\ \begin{aligned} ∥r^{'} (t)∥ = \sqrt{\sin^{2} t + \cos^{2} t + 1} \\ = \sqrt{2} \end{aligned} \\ T (t) = \frac{r^{'} (t)}{∥r^{'} (t)∥} \\ = \frac{⟨ - \sin t, \cos t, 1 ⟩}{\sqrt{2}} \\ At t = π, T (t) = T (π) = \frac{⟨ - \sin π, \cos π, 1 ⟩}{\sqrt{2}} \\ = \frac{⟨ 0, - 1, 1 ⟩}{\sqrt{2}} \\ = <0, \frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \end{array}$

$Thus the unit tangent vector to r (t) at t = π is \begin{array}{l} T (π) = <0, \frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \\ T^{'} (t) = \frac{1}{\sqrt{2}} ⟨ - \cos t, - \sin t, 0 ⟩ \\ ∥T^{'} (t)∥ = \sqrt{\frac{1}{2} \cos^{2} t + \frac{1}{2} \sin^{2} t} \\ = \frac{1}{\sqrt{2}} \\ N (t) = \frac{T^{'} (t)}{∥T^{'} (t)∥} = \frac{\frac{1}{\sqrt{2}} ⟨ - \cos t, - \sin t, 0 ⟩}{\frac{- 1}{\sqrt{2}}} \\ = ⟨ - \cos t, - \sin t, 0 ⟩ \\ At t = π, N (t) = N (π) = ⟨ - \cos π, - \sin π, 0 ⟩ \\ = ⟨ 1, 0, 0 ⟩ \\ Thus the principal unit normal vector to r (t) at t = π is ⟨ 1, 0, 0 ⟩ \\ B (π) = T (π) \times N (π) \\ \begin{array}{l} = |\begin{array}{ccc} i & j & k \\ 0 & \frac{- \sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ 1 & 0 & 0 \end{array}| \\ = i (0) - j (\frac{- \sqrt{2}}{2}) + k (\frac{\sqrt{2}}{2}) \\ = <0, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}> \end{array} \end{array}$

Step2. Rectifying plane

$The plane determined by the vectors T (t_{0}) and B (t_{0}) containing the point r (t_{0}) is called the rectifying plane for C at r (t_{0}) . Thus the equation of the rectifying plane to r (t) at t = π is (T (π) \times B (π)) \cdot ⟨ x - x (π), y - y (π), z - z (π) ⟩ = 0 First computing T (π) \times B (π) :$

$\begin{aligned} = |\begin{array}{ccc} i & j & k \\ 0 & \frac{- \sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array}| \\ = |\begin{array}{ccc} i & j & k \\ 0 & \frac{- \sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array}| \end{aligned}$

$Evaluating \begin{array}{l} (T (π) \times B (π)) \cdot ⟨ x - x (π), y - y (π), z - z (π) ⟩ = 0 \\ ⟨ - 1, 0, 0 ⟩ \cdot ⟨ x + 1, y, z - π ⟩ = 0 \\ \Rightarrow - 1 (x + 1) = 0 \\ \Rightarrow x + 1 = 0 \\ Thus the equation of the rectifying plane to r (t) at t = π is \\ x + 1 = 0 \end{array}$

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Let C be the graph of a vector-valued function r. The plane determined by the vectors T(t0) and B(t0) and containing the point r(t0) is called the rectifying plane for C at r(t0). Find the equation of the rectifying plane to the helix determined by r(t)=(cost,sint,t) when t = π.

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