$r\left(t\right)=\left(e^t\sin t,e^t\cos t,e^t\right)$

Consider

The goal is to determine the tangential and normal components of acceleration. $r\left(t\right)$.

The tangential and normal components of Acceleration

$r\left(t\right)$ is a twice - differentiable vector function with$r^{\text{'}}\left(t\right)=v\left(t\right)$and $r^{\text{'}\text{'}}\left(t\right)=a\left(t\right)$

The tangential component of acceleration, $a_T=\frac{v·a}{\Vert v\Vert }$. The normal component of acceleration,

$a_N=\frac{\Vert v\times a\Vert }{\Vert v\Vert }$

$v\left(t\right)=r^{\text{'}}\left(t\right)$

$a\left(t\right)=r^{n\text{'}}\left(t\right)$

$=\left\{\begin{array}{l}-e^{\text{'}}\sin t+e^{\text{'}}\cos t+e^{\text{'}}\cos t+e^{\text{'}}\sin t-e^{\text{'}}\cos t-e^{\text{'}}\sin t\\ -e^{\text{'}}\sin t+e^{\text{'}}\cos t,e^{\text{'}}\end{array}\right)$

$a\left(t\right)=r^{\text{'}\text{'}}\left(t\right)$

$a\left(t\right)=r^{t\text{'}}\left(t\right)$

$\Vert v\Vert =\Vert \left(e^t\cos t+e^t\sin t,-e^t\sin t+e^t\cos t,e^t\right)$

$=\sqrt{{\left(e^t\cos t+e^{\text{'}}\sin t\right)}^2+{\left(-e^t\sin t+e^t\cos t\right)}^2+e^{2t}}$

$=\sqrt{e^{2t}{\cos }^{2}t+e^{2t}{\sin }^{2}t+e^{2t}{\sin }^{2}t+e^{2t}{\cos }^{2}t+e^{2t}}$

$=\sqrt{2e^{2t}\left({\cos }^{2}t+{\sin }^{2}t\right)+e^{2t}}$

$=\sqrt{3e^{2t}}$

$=\sqrt{3}{e}^t$

$v·a=v\left(t\right)·a\left(t\right)$

$=2e^t\cos t\left(e^t\cos t+e^t\sin t\right)-2e^t\sin t\left(-e^t\sin t+e^t\cos t\right)+e^{2t}$

$=2e^{2t}{\cos }^{2}t+2e^{2t}\sin t\cos t+2e^{2t}{\sin }^{2}t-2e^{2t}\sin t\cos t+e^{2t}$

$=2e^{2t}\left({\cos }^{2}t+{\sin }^{2}t\right)+e^{2t}$

$=3e^{2t}$

$v\times a=v\left(t\right)\times a\left(t\right)$

$=\left|\begin{array}{c}ijk\\ e^{\text{'}}\cos t+e^{\text{'}}\sin t-e^{\text{'}}\sin t+e^{\text{'}}\cos t{e}^t\\ 2e^t\cos t-2e^t\sin t{e}^{\text{'}}\end{array}\right|$

$=i\left(e^{\prime}\left(-e^{\prime} \sin t+e^{\prime} \cos t\right)+e^{\prime}\left(2 e^{\prime} \sin t\right)\right)$

$-j\left(e^{\prime}\left(e^{\prime} \cos t+e^{\prime} \sin t\right)-e^{\prime}\left(2 e^{t} \cos t\right)\right)$

$+k\left(-2 e^{\prime} \sin t\left(e^{\prime} \cos t+e^{\prime} \sin t\right)-2 e^{\prime} \cos t\left(-e^{\prime} \sin t+e^{\prime} \cos t\right)\right)$

$=i\left(e^{2 t} \sin t+e^{2 t} \cos t\right)-j\left(e^{2 t} \sin t-e^{2 t} \cos t\right)+k\left(-2 e^{2 t} \sin ^{2} t-2 e^{2 t} \cos ^{2} t\right)$

$=i\left(e^{2 t}(\sin t+\cos t)\right)-j\left(e^{2 t}(\sin t-\cos t)\right)-2 e^{2 t} k$

$\|v \times a\|=\sqrt{\left(e^{2 t}(\sin t+\cos t)\right)^{2}+\left(-e^{2 t}(\sin t-\cos t)\right)^{2}+4 e^{4 t}}$

$=\sqrt{e^{4 t}\left(\sin ^{2} t+\cos ^{2} t+\sin ^{2} t+\cos ^{2} t\right)+4 e^{4 t}}$

$=\sqrt{2 e^{4 t}+4 e^{4 t}}$

$=\sqrt{6 a^{4 !}}$