Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve.
Verify that T= N = 1 and T. N = 0.
(a) r(t) = (10 cos t^2, 10 sin t^2) for 0 < t < 2 pi
(b) r(t) = (t^2/2 , 1-6t, -9)

a.

b.

Step-by-step explanation:

Remember that for any curve        

The tangent vector is given by

And the normal vector is given by

a.

For this case, using the chain rule

And also remember that

Therefore

Similarly, using the quotient rule and the chain rule

And also

Therefore

Notice that

1.  

2.  

b.

Simlarly

and

Therefore

Then

and also

And since

step-by-step explan mxfcg'k; uafwgzfgjhklhgation:

c. 1,233 miles