Tangential component of acceleration for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle 3 \cos{\left(5 t \right)}, 3 \sin{\left(5 t \right)}, 8 t\right\rangle$$$

Find the tangential component of acceleration for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle 3 \cos{\left(5 t \right)}, 3 \sin{\left(5 t \right)}, 8 t\right\rangle$$$.

Find the derivative of $$$\mathbf{\vec{r}\left(t\right)}$$$: $$$\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle - 15 \sin{\left(5 t \right)}, 15 \cos{\left(5 t \right)}, 8\right\rangle$$$ (for steps, see derivative calculator).

Find the magnitude of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}$$$: $$$\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert} = 17$$$ (for steps, see magnitude calculator).

Find the derivative of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}$$$: $$$\mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle - 75 \cos{\left(5 t \right)}, - 75 \sin{\left(5 t \right)}, 0\right\rangle$$$ (for steps, see derivative calculator).

Find the dot product: $$$\mathbf{\vec{r}^{\prime}\left(t\right)}\cdot \mathbf{\vec{r}^{\prime\prime}\left(t\right)} = 0$$$ (for steps, see dot product calculator).

Finally, the tangential component of acceleration is $$$a_T\left(t\right) = \frac{\mathbf{\vec{r}^{\prime}\left(t\right)}\cdot \mathbf{\vec{r}^{\prime\prime}\left(t\right)}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert}} = 0.$$$

The tangential component of acceleration is $$$a_T\left(t\right) = 0$$$A.