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

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

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

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

### 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).

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

Find the unit vector: $\mathbf{\vec{T}\left(t\right)} = \left\langle \frac{\cos{\left(t \right)}}{3}, - \frac{\sin{\left(t \right)}}{3}, \frac{2 \sqrt{2}}{3}\right\rangle$ (for steps, see unit vector calculator).

The unit tangent vector is $\mathbf{\vec{T}\left(t\right)} = \left\langle \frac{\cos{\left(t \right)}}{3}, - \frac{\sin{\left(t \right)}}{3}, \frac{2 \sqrt{2}}{3}\right\rangle$A.