Final answer:
To find the curvature of a curve defined by the position vector, we need to calculate the first and second derivatives of the vector and use the formula: curvature = ||r'(t) x r''(t)|| / ||r'(t)||^3
Step-by-step explanation:
The formula to find the curvature of a curve in space is given by: r(t) = x(t)i + y(t)j + z(t)k
To find the curvature, we need to calculate the first and second derivatives of the position vector R(t) = 8t i + 5 sin(t) j + 5 cos(t) k. Let's denote the derivatives as r'(t) and r''(t) respectively. Then, the curvature is given by: curvature = ||r'(t) x r''(t)|| / ||r'(t)||^3
By substituting the values from the position vector, we can calculate the derivatives and find the curvature.