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Unit Normal Vector Calculator

Calculate unit normal vectors step by step

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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Your Input

Find the principal unit normal vector for r⃗(t)=⟨sin⁡(t),cos⁡(t),22t⟩\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 T⃗(t)\mathbf{\vec{T}\left(t\right)} and then normalize it (find the unit vector).

Find the unit tangent vector: T⃗(t)=⟨cos⁡(t)3,−sin⁡(t)3,223⟩\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).

T⃗′(t)=⟨−sin⁡(t)3,−cos⁡(t)3,0⟩\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: N⃗(t)=⟨−sin⁡(t),−cos⁡(t),0⟩\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).

Answer

The principal unit normal vector is N⃗(t)=⟨−sin⁡(t),−cos⁡(t),0⟩\mathbf{\vec{N}\left(t\right)} = \left\langle - \sin{\left(t \right)}, - \cos{\left(t \right)}, 0\right\rangleA.