Question

Use the alternative curvature formula k=|a×v|/|v|³ to find the curvature of the following parameterized curve:

r(t) = (7 cos t, √2 sint,2 cost)

Answer 1

The curvature (k) of the parameterized curve
`\(r(t) = (7 \cos t, √(2) \sin t, 2 \cos t)\)` is given by
`\(k = \frac{|\mathbf{a} * \mathbf{v}|}{|\mathbf{v}|^3}\)`, where \(\mathbf{a}\) is the acceleration vector and
`\(\mathbf{v}\)` is the velocity vector.

Step-by-step explanation:

To find the curvature using the alternative formula, we need to calculate the acceleration vector
`\(\mathbf{a}\)` and the velocity vector v first. The velocity vector is the derivative of the position vector, and the acceleration vector is the derivative of the velocity vector. For the given parameterized curve
`\(r(t) = (7 \cos t, √(2) \sin t, 2 \cos t)\)`, we find that the velocity vector
`\(\mathbf{v}\)` is the derivative of
`\(r(t)\)`, and the acceleration vector
`\(\mathbf{a}\)` is the derivative of
`\(\mathbf{v}\)`.

After finding
`\(\mathbf{a}\) and \(\mathbf{v}\)`, we substitute these values into the curvature formula
`\(k = \frac{|\mathbf{a} * \mathbf{v}|}{|\mathbf{v}|^3}\)`. This involves finding the cross product of \(\mathbf{a}\) and v, taking the magnitude, and dividing by the cube of the magnitude of v.

In conclusion, by applying the alternative curvature formula to the parameterized curve
`\(r(t) = (7 \cos t, √(2) \sin t, 2 \cos t)\)`, we can determine the curvature at any given point on the curve. The process involves finding the acceleration and velocity vectors and applying the formula to obtain the curvature value.