Unit tangent vector for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle e^{2 t}, e^{-7}\right\rangle$$$ at $$$t = 0$$$

The calculator will find the unit tangent vector to $$$\mathbf{\vec{r}\left(t\right)} = \left\langle e^{2 t}, e^{-7}\right\rangle$$$ at $$$t = 0$$$, with steps shown.

Related calculators: Unit Normal Vector Calculator, Unit Binormal Vector Calculator

$$$\langle$$$ $$$\rangle$$$
Leave empty if you don't need the vector at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

Find the unit tangent vector for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle e^{2 t}, e^{-7}\right\rangle$$$ at $$$t = 0$$$.


To find the unit tangent vector, we need to find the derivative of $$$\mathbf{\vec{r}\left(t\right)}$$$ (the tangent vector) and then normalize it (find the unit vector).

$$$\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle 2 e^{2 t}, 0\right\rangle$$$ (for steps, see derivative calculator).

Find the unit vector: $$$\mathbf{\vec{T}\left(t\right)} = \left\langle 1, 0\right\rangle$$$ (for steps, see unit vector calculator).

Now, find the vector at $$$t = 0$$$.

$$$\mathbf{\vec{T}\left(0\right)} = \left\langle 1, 0\right\rangle$$$


The unit tangent vector is $$$\mathbf{\vec{T}\left(t\right)} = \left\langle 1, 0\right\rangle$$$A.

$$$\mathbf{\vec{T}\left(0\right)} = \left\langle 1, 0\right\rangle$$$A