Question

At any point of the path x=3cos⁡t,y=3sin⁡t,z=4t, find the Tangent vector . n

Answer 1

$$\vec{r}(t)=(3\cos t, 3\sin t, 4t)$$

$$\vec{r'}(t)=(-3\sin t, 3\cos t, 4)$$

$$|\vec{r'}(t)|=\sqrt{(-3\sin t)^2+(3\cos t)^2+(4)^2}=5$$

$$\vec{T}(t)=\dfrac{\vec{r'}(t)}{|\vec{r'}(t)|}=(-\dfrac{3}{5}\sin t, \dfrac{3}{5}\cos t, \dfrac{4}{5})$$

$$|\vec{T}(t)|=\sqrt{(-\dfrac{3}{5}\sin t)^2+(\dfrac{3}{5}\cos t)^2+(\dfrac{4}{5})^2}=1$$

$$\vec{N}(t)=\dfrac{\vec{T'}(t)}{|\vec{T'}(t)|}$$

$$\vec{T'}(t)=(-\dfrac{3}{5}\cos t, -\dfrac{3}{5}\sin t, 0)$$

$$|\vec{T'}(t)|=\sqrt{(-\dfrac{3}{5}\cos t)^2+(-\dfrac{3}{5}\sin t)^2+(0)^2}=\dfrac{3}{5}$$

$$\vec{N}(t)=(-\cos t, -\sin t, 0)$$