# Torsion Calculator

## Calculate torsion step by step

The calculator will find the torsion of the given vector-valued function at the given point, with steps shown.

Related calculator: Curvature Calculator

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Leave empty if you don't need the torsion at a specific point.

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Find the torsion of $\mathbf{\vec{r}\left(t\right)} = \left\langle t^{2}, t^{3}, t\right\rangle$.

### Solution

Find the derivative of $\mathbf{\vec{r}\left(t\right)}$: $\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle 2 t, 3 t^{2}, 1\right\rangle$ (for steps, see derivative calculator).

Find the derivative of $\mathbf{\vec{r}^{\prime}\left(t\right)}$: $\mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle 2, 6 t, 0\right\rangle$ (for steps, see derivative calculator).

Find the cross product: $\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle - 6 t, 2, 6 t^{2}\right\rangle$ (for steps, see cross product calculator).

Find the magnitude of $\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}$: $\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert} = 2 \sqrt{9 t^{4} + 9 t^{2} + 1}$ (for steps, see magnitude calculator).

Find the derivative of $\mathbf{\vec{r}^{\prime\prime}\left(t\right)}$: $\mathbf{\vec{r}^{\prime\prime\prime}\left(t\right)} = \left\langle 0, 6, 0\right\rangle$ (for steps, see derivative calculator).

Find the dot product: $\left(\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right)\cdot \mathbf{\vec{r}^{\prime\prime\prime}\left(t\right)} = 12$ (for steps, see dot product calculator).

Finally, the torsion is $\tau\left(t\right) = \frac{\left(\mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right)\cdot \mathbf{\vec{r}^{\prime\prime\prime}\left(t\right)}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\times \mathbf{\vec{r}^{\prime\prime}\left(t\right)}\right\rvert}^{2}} = \frac{3}{9 t^{4} + 9 t^{2} + 1}.$

The torsion is $\tau\left(t\right) = \frac{3}{9 t^{4} + 9 t^{2} + 1}$A.