Unit tangent vector for r⃗(t) = ⟨e^(2*t), e^(-7)⟩ 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.

Your Input

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

Solution

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 $$$t = 0$$$.

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

Answer

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

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

Your Input

Solution

Answer