Question

Find the curvature for the circle r'(t) = 4sin(t)i - 4cos(t)j.

Which formula would you use to measure curvature κ (kappa)?

κ = ||r'(t) x T'(t)||
κ = ||r'(t) x T'(t)||
κ = ||r'(t) · T'(t)||
κ = ||r(t) · T'(t)||

Answer 1

To find the curvature for the circle represented by the vector function r'(t) = 4sin(t)i - 4cos(t)j, use the formula κ = ||r'(t) x T'(t)||.

Step-by-step explanation:

To find the curvature for the circle represented by the vector function r'(t) = 4sin(t)i - 4cos(t)j, we can use the formula κ = ||r'(t) x T'(t)||, where r'(t) is the first derivative of the vector function and T'(t) is the derivative of the unit tangent vector. This formula measures the rate at which the tangent vector changes with respect to the position vector, giving us the rate of change of the direction of the curve. Therefore, the correct formula to measure the curvature κ (kappa) is κ = ||r'(t) x T'(t)||.