?[T] Consider \(\mathbf{r}(t)=\langle\cos t, \sin t, 2 t\rangle\) the position vector of

Chapter 2, Problem 182

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[T] Consider \(\mathbf{r}(t)=\langle\cos t, \sin t, 2 t\rangle\) the position vector of a particle at time \(t \in[0,30]\), where the components of r are expressed in centimeters and time in seconds. Let \(\overrightarrow{O P}\) be the position vector of the particle after 1 sec.

a. Show that all vectors \(\overrightarrow{P Q}\) , where Q(x, y, z) is an arbitrary point, orthogonal to the instantaneous velocity vector v(1) of the particle after 1 sec, can be expressed as \(\overrightarrow{P Q}=\langle x-\cos 1, y-\sin 1, z-2\rangle\) , where x sin 1 − y cos 1 − 2z + 4 = 0. The set of point Q describes a plane called the normal plane to the path of the particle at point P.

b. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point P along with the path of the particle.

Text Transcription:

r(t) = langle cos t, sin t, 2t rangle

t in [0,30]

overrightarrow OP

overrightarrow PQ

overrightarrow PQ = langle x - cos 1, y - sin 1, z-2 rangle