Question

Find the curvature of r(t)=3cos t i+2sin t j+t k. Which formula would you use to measure curvature κ (kappa):

Answer 1

To find the curvature of r(t) = 3cos(t)i + 2sin(t)j + tk, we need to find the derivatives of r(t) and T(t) using the formula κ = ||T'(t)|| / ||r'(t)||.

Step-by-step explanation:

The formula used to measure curvature κ (kappa) is given by the equation κ = ||T'(t)|| / ||r'(t)||, where T(t) is the unit tangent vector and r(t) is the position vector.

To find the curvature of r(t) = 3cos(t)i + 2sin(t)j + tk, we need to find the derivatives of r(t) and T(t).

Let's start by finding the derivatives:

1. Find r'(t) by taking the derivative of each component: r'(t) = -3sin(t)i + 2cos(t)j + k
2. To find T(t), we need to normalize r'(t) by dividing it by its magnitude: T(t) = (1/√(14))*(-3sin(t)i + 2cos(t)j + k)
3. Now, find T'(t) by taking the derivative of T(t): T'(t) = (1/√(14))*(-3cos(t)i - 2sin(t)j)

Finally, substitute the values of T'(t) and r'(t) into the formula κ = ||T'(t)|| / ||r'(t)|| to calculate the curvature.