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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.

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

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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).

Answer

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.