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