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

Chapter 2, Problem 241

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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. Determine unit vector B(t) (called the binormal unit vector) that has the direction of cross product vector \(\mathbf{v}(t) \times \mathbf{a}(t)\), where v(t) and a(t) are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after t seconds.

b. Use a CAS to visualize vectors v(1), a(1), and B(1) as vectors starting 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

v(t) times a(t)