Use the alternative curvature formula $$ \kappa=\frac{|a imes v|}{|v|^{3}} $$ to find the curvature of the following parameterized curve. $$ r(t)=\langle 5 \cos t, \sqrt{5} \sin t, 2 \cos t angle $$ $$ \kappa=\square $$

Question

Use the alternative curvature formula $$ \kappa=\frac{|a 	imes v|}{|v|^{3}} $$ to find the curvature of the following parameterized curve. $$ r(t)=\langle 5 \cos t, \sqrt{5} \sin t, 2 \cos tangle $$ $$ \kappa=\square $$

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To find the curvature $$ \kappa $$ of the parameterized curve given by $$ r(t) = \langle 5 \cos t, \sqrt{5} \sin t, 2 \cos t \rangle $$, we will follow these steps:

1. **Calculate the velocity vector $$ v(t) $$** by differentiating $$ r(t) $$ with respect to $$ t $$.
2. **Calculate the acceleration vector $$ a(t) $$** by differentiating $$ v(t) $$ with respect to $$ t $$.
3. **Compute the cross product $$ a \times v $$**.
4. **Calculate the magnitudes $$ |a \times v| $$ and $$ |v| $$**.
5. **Substitute these values into the curvature formula $$ \kappa = \frac{|a \times v|}{|v|^3} $$**.

Let's start with step 1 and calculate the velocity vector $$ v(t) $$:

$$
v(t) = \frac{dr}{dt} = \left\langle \frac{d}{dt}(5 \cos t), \frac{d}{dt}(\sqrt{5} \sin t), \frac{d}{dt}(2 \cos t) \right\rangle
$$

Calculating the derivatives:

$$
v(t) = \left\langle -5 \sin t, \sqrt{5} \cos t, -2 \sin t \right\rangle
$$

Next, we will calculate the acceleration vector $$ a(t) $$:

$$
a(t) = \frac{dv}{dt} = \left\langle \frac{d}{dt}(-5 \sin t), \frac{d}{dt}(\sqrt{5} \cos t), \frac{d}{dt}(-2 \sin t) \right\rangle
$$

Calculating the derivatives:

$$
a(t) = \left\langle -5 \cos t, -\sqrt{5} \sin t, -2 \cos t \right\rangle
$$

Now, we will compute the cross product $$ a \times v $$:

$$
a \times v = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
-5 \cos t & -\sqrt{5} \sin t & -2 \cos t \
-5 \sin t & \sqrt{5} \cos t & -2 \sin t
\end{vmatrix}
$$

Calculating the determinant:

$$
a \times v = \mathbf{i} \left( (-\sqrt{5} \sin t)(-2 \sin t) - (-2 \cos t)(\sqrt{5} \cos t) \right) - \mathbf{j} \left( (-5 \cos t)(-2 \sin t) - (-2 \cos t)(-5 \sin t) \right) + \mathbf{k} \left( (-5 \cos t)(\sqrt{5} \cos t) - (-5 \sin t)(-\sqrt{5} \sin t) \right)
$$

Calculating each component:

1. For $$ \mathbf{i} $$:
   $$
   2\sqrt{5} \sin^2 t - 2\sqrt{5} \cos^2 t = 2\sqrt{5} (\sin^2 t - \cos^2 t)
   $$

2. For $$ \mathbf{j} $$:
   $$
   10 \cos t \sin t - 10 \sin t \cos t = 0
   $$

3. For $$ \mathbf{k} $$:
   $$
   -5\sqrt{5} \cos^2 t + 5\sqrt{5} \sin^2 t = 5\sqrt{5} (\sin^2 t - \cos^2 t)
   $$

Thus, we have:

$$
a \times v = \left\langle 2\sqrt{5} (\sin^2 t - \cos^2 t), 0, 5\sqrt{5} (\sin^2 t - \cos^2 t) \right\rangle
$$

Next, we calculate the magnitude $$ |a \times v| $$:

$$
|a \times v| = \sqrt{(2\sqrt{5} (\sin^2 t - \cos^2 t))^2 + 0^2 + (5\sqrt{5} (\sin^2 t - \cos^2 t))^2}
$$

Calculating this:

$$
|a \times v| = \sqrt{(4 \cdot 5 (\sin^2 t - \cos^2 t)^2) + (25 \cdot 5 (\sin^2 t - \cos^2 t)^2)} = \sqrt{(4 + 25) \cdot 5 (\sin^2 t - \cos^2 t)^2} = \sqrt{145} |\sin^2 t - \cos^2 t|
$$

Now, we calculate the magnitude $$ |v| $$:

$$
|v| = \sqrt{(-5 \sin t)^2 + (\sqrt{5} \cos t)^2 + (-2 \sin t)^2} = \sqrt{25 \sin^2 t + 5 \cos^2 t + 4 \sin^2 t} = \sqrt{29 \sin^2 t + 5 \cos^2 t}
$$

Finally, we substitute these values into the curvature formula:

$$
\kappa = \frac{|a \times v|}{|v|^3} = \frac{\sqrt{145} |\sin^2 t - \cos^2 t|}{(29 \sin^2 t + 5 \cos^2 t)^{3/2}}
$$

Thus, the curvature $$ \kappa $$ is:

$$
\kappa = \frac{\sqrt{145} |\sin^2 t - \cos^2 t|}{(29 \sin^2 t + 5 \cos^2 t)^{3/2}}
$$
The curvature $$ \kappa $$ of the given curve is $$ \frac{\sqrt{145} |\sin^2 t - \cos^2 t|}{(29 \sin^2 t + 5 \cos^2 t)^{3/2}} $$.

Use the alternative curvature formula \( \kappa=\frac{|a \times v|}{|v|^{3}} \) to find the curvature of the following parameterized curve. \( r(t)=\langle 5 \cos t, \sqrt{5} \sin t, 2 \cos t\rangle \) \( \kappa=\square \)

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