MHB Find the curvature of the vector

[carl123]

Find the curvature.

r(t) = 3t i + 5t j + (6 + t²) k

κ(t) =

This is what i have so far:

derivative of r(t) = 3i + 5j + 2k

I know the next step is to find the cross product of r(t) and r^'(t) but I'm not sure how to go about it, especially how to make use of the k vector of r(t)

 

[Ackbach]

Find the curvature.

r(t) = 3t i + 5t j + (6 + t²) k

κ(t) =

This is what i have so far:

derivative of r(t) = 3i + 5j + 2k

This should be $... + 2t \mathbf{k}$.

I know the next step is to find the cross product of r(t) and r^'(t) but I'm not sure how to go about it, especially how to make use of the k vector of r(t)

I would do
$$\frac{|\dot{\mathbf{r}}(t) \times \ddot{\mathbf{r}}(t)|}{|\dot{\mathbf{r}}(t)|^3}.$$

In general,
$$\mathbf{A}\times\mathbf{B}=\left|\begin{matrix}
\mathbf{i} &\mathbf{j} &\mathbf{k} \\
A_x &A_y &A_z \\
B_x &B_y &B_z
\end{matrix} \right|.$$
Can you continue?