# Unit Normal Vector Calculator

The calculator will find the principal unit normal vector to the vector-valued function at the given point, with steps shown.

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

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Find the principal unit normal 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 principal unit normal vector, we need to find the derivative of the unit tangent vector $\mathbf{\vec{T}\left(t\right)}$ and then normalize it (find the unit vector).

Find the unit tangent 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 tangent vector calculator).

$\mathbf{\vec{T}^{\prime}\left(t\right)} = \left\langle - \frac{\sin{\left(t \right)}}{3}, - \frac{\cos{\left(t \right)}}{3}, 0\right\rangle$ (for steps, see derivative calculator).

Find the unit vector: $\mathbf{\vec{N}\left(t\right)} = \left\langle - \sin{\left(t \right)}, - \cos{\left(t \right)}, 0\right\rangle$ (for steps, see unit vector calculator).

The principal unit normal vector is $\mathbf{\vec{N}\left(t\right)} = \left\langle - \sin{\left(t \right)}, - \cos{\left(t \right)}, 0\right\rangle$A.