Question

My teacher admitted that my book didn't explain how to do this well enough..I need help..Any ideas? Award At End. Given r(t), find the following: r(t) = < t^2, sin(t) - tcos(t), cos(t) + tsin(t) > Write a(t) in terms of its tangential and normal components.

Answer 1

Differentiate once to find the velocity v(t) = r'(t).
Differentiate the velocity vector to find acceleration. a(t) = v'(t).

Find the unit tangent and normal vectors:
When you differentiate r(t) once, you have an equation that gives you the slope of the tangent to the curve as a function of t. So the velocity vector also happens to be the tangent vector. Therefore, unit tangent vector = v(t) / ||v(t)||.

Do a dot product of a(t) with unit tangent vector to find the tangential component of the acceleration.

Find the principal unit normal vector using the formula: N(t) = T'(t) / ||T'(t)|| where T(t) is the unit tangent vector found earlier and T'(t) is its derivative.

Answer 2

a(t) dot product with unit normal vector will give you the normal component of the acceleration.

This link may explain further: http://ltcconline.net/greenl/courses/202/vectorFunctions/tannorm.htm