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Principal unit normal vector for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle t, \frac{t^{2}}{2}, t^{2}\right\rangle$$$
Related calculators: Unit Tangent Vector Calculator, Unit Binormal Vector Calculator
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Find the principal unit normal vector for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle t, \frac{t^{2}}{2}, t^{2}\right\rangle$$$.
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: $$$\mathbf{\vec{T}\left(t\right)} = \left\langle \frac{1}{\sqrt{5 t^{2} + 1}}, \frac{t}{\sqrt{5 t^{2} + 1}}, \frac{2 t}{\sqrt{5 t^{2} + 1}}\right\rangle$$$ (for steps, see unit tangent vector calculator).
$$$\mathbf{\vec{T}^{\prime}\left(t\right)} = \left\langle - \frac{5 t}{\left(5 t^{2} + 1\right)^{\frac{3}{2}}}, \frac{1}{\left(5 t^{2} + 1\right)^{\frac{3}{2}}}, \frac{2}{\left(5 t^{2} + 1\right)^{\frac{3}{2}}}\right\rangle$$$ (for steps, see derivative calculator).
Find the unit vector: $$$\mathbf{\vec{N}\left(t\right)} = \left\langle - \frac{\sqrt{5} t}{\sqrt{5 t^{2} + 1}}, \frac{\sqrt{5}}{5 \sqrt{5 t^{2} + 1}}, \frac{2 \sqrt{5}}{5 \sqrt{5 t^{2} + 1}}\right\rangle$$$ (for steps, see unit vector calculator).
Answer
The principal unit normal vector is $$$\mathbf{\vec{N}\left(t\right)} = \left\langle - \frac{\sqrt{5} t}{\sqrt{5 t^{2} + 1}}, \frac{\sqrt{5}}{5 \sqrt{5 t^{2} + 1}}, \frac{2 \sqrt{5}}{5 \sqrt{5 t^{2} + 1}}\right\rangle.$$$A