# Unit tangent vector for $\mathbf{\vec{r}\left(t\right)} = \left\langle t^{2}, 2 t\right\rangle$

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

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

### 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).

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

Find the unit vector: $\mathbf{\vec{T}\left(t\right)} = \left\langle \frac{t}{\sqrt{t^{2} + 1}}, \frac{1}{\sqrt{t^{2} + 1}}\right\rangle$ (for steps, see unit vector calculator).

The unit tangent vector is $\mathbf{\vec{T}\left(t\right)} = \left\langle \frac{t}{\sqrt{t^{2} + 1}}, \frac{1}{\sqrt{t^{2} + 1}}\right\rangle$A.