Question

Find the curvature. r(t) = 3t i + 9t j + (1 + t2) k

Answer 1

To find the curvature of the given position vector, we need to find its first derivative and calculate its magnitude. Then, using the formula for curvature, we can calculate the curvature by dividing the magnitude of the derivative by the magnitude of the first derivative.

Step-by-step explanation:

The formula for finding the curvature of a curve is given by:

κ = |r'(t)| / ||r'(t)||

In this case, the position vector r(t) = 3t i + 9t j + (1 + t2) k. To find the curvature, we need to find the first derivative of r(t) and calculate its magnitude.

1. Find the derivative of r(t): Differentiate each component of the position vector to get:
   r'(t) = 3i + 9j + 2t k.
2. Calculate the magnitude of r'(t): The magnitude is given by:
   ||r'(t)|| = sqrt(32 + 92 + (2t)2) = sqrt(90 + 4t2).
3. Calculate the curvature: The curvature is given by the formula:
   κ = |r'(t)| / ||r'(t)||. Substitute the values of |r'(t)| and ||r'(t)|| to get the final result.