Question

Use the alternative curvature formula κ=|v X a|/|v|^3 to find the curvature of the following parameterized curve. r(t) = (5e^t cost. 5e^t sint, 3e^t).

Answer 1

The curvature of the curve r(t) = (5e^t cos t, 5e^t sin t, 3e^t) can be calculated using the formula κ = |v × a|/|v|^3 by first finding the derivatives to obtain the velocity and acceleration vectors, their cross product, and the magnitude of these vectors.

Step-by-step explanation:

To find the curvature of the parameterized curve r(t) = (5e^t cos t, 5e^t sin t, 3e^t), we use the alternative curvature formula: κ = |v × a|/|v|^3. First, we need to determine the velocity (v) and acceleration (a) vectors of the curve.

The velocity vector v(t) is the derivative of the position vector r(t) with respect to time, and the acceleration vector a(t) is the derivative of the velocity vector v(t) with respect to time. After finding these vectors, we'll calculate the cross product of v and a, obtain its magnitude, and determine the magnitude of the velocity to the third power. Finally, dividing the magnitude of the cross product by the cube of the velocity magnitude will give us the curvature.

Note: To effectively assist with the calculation, it is important to work through the derivatives and cross product step by step to ensure accuracy.