Español | Português
  1. Home
  2. Calculators
  3. Calculators: Calculus III
  4. Calculus Calculator

Unit tangent vector for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle \cos{\left(t \right)}, \sqrt{3} t, \sin{\left(t \right)}\right\rangle$$$

The calculator will find the unit tangent vector to $$$\mathbf{\vec{r}\left(t\right)} = \left\langle \cos{\left(t \right)}, \sqrt{3} t, \sin{\left(t \right)}\right\rangle$$$, with steps shown.

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

$$$\langle$$$ $$$\rangle$$$
Comma-separated.
Leave empty if you don't need the vector at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

Find the unit tangent vector for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle \cos{\left(t \right)}, \sqrt{3} t, \sin{\left(t \right)}\right\rangle$$$.

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).

$$$\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle - \sin{\left(t \right)}, \sqrt{3}, \cos{\left(t \right)}\right\rangle$$$ (for steps, see derivative calculator).

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

Answer

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