Unit Tangent Vector Calculator

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

Enter a vector-valued function:

`mathbf{vec{r}(t)}=` (, , )
If you don't have the third coordinate, set it to `0`.
Calculate at `t=`
Leave empty, if you don't need the unit tangent vector at a specific point.

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