# Unit Tangent Vector Calculator

The calculator will find the unit tangent vector of a vector-valued function at the given point, with steps shown.

## Solution

**Your input: find the unit tangent vector for $$$\mathbf{\vec{r}(t)}=\left(\sin{\left(t \right)}, \cos{\left(t \right)}, 7\right)$$$**

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

$$$\mathbf{\vec{r}^{\prime}(t)}=\left(\cos{\left(t \right)}, - \sin{\left(t \right)}, 0\right)$$$

*Note*. For steps in finding derivatives, see derivative calculator.

Find the norm (length) of the vector: $$$\lVert\mathbf{\vec{r}^{\prime}(t)}\rVert=\sqrt{\left(\cos{\left(t \right)}\right)^2+\left(- \sin{\left(t \right)}\right)^2+\left(0\right)^2}=1$$$

Finally, the unit tangent vector is $$$\mathbf{\vec{T}(t)}=\frac{\mathbf{\vec{r}^{\prime}(t)}}{\lVert\mathbf{\vec{r}^{\prime}(t)}\rVert}$$$

$$$\mathbf{\vec{T}(t)}=\frac{\left(\cos{\left(t \right)}, - \sin{\left(t \right)}, 0\right)}{1}=\left(\cos{\left(t \right)}, - \sin{\left(t \right)}, 0\right)$$$

**Answer: $$$\mathbf{\vec{T}(t)}=\left(\cos{\left(t \right)}, - \sin{\left(t \right)}, 0\right)$$$**