Find the unit tangent vector and the principal unit normal vector for the follow

Question

Find the unit tangent vector and the principal unit normal vector for the following parameterized curve

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Find the unit tangent vector T rightarrow and the principal unit normal vector N rightarrow for the following parameterized curve. r rightarrow (t) = 7 sin t, 7 cos t, 24t T rightarrow = (Simplify your answers. Write the trigonometric functions in the same format as in the question above.) N rightarrow = (Simplify your answers. Write the trigonometric functions in the same format as in the question above.)

Explanation / Answer

Since r(t) = <7sint,7cost,24t>,

r'(t) = <7cost,-7sint,24>.

||r'(t)|| =[49([cost]^2+[sint]^2)+24^2]^(1/2)

=25

we have T(t) = r'(t)/||r'(t)|| =1/25 * <7cost,-7sint,24>

T'(t) = 1/25 * <-7sint,-7cost,0>

||T'(t)|| = 7/25

==> N(t) = T'(t)/||T'(t)|| =[ 1/25 * <-7sint,-7cost,0> ]/[7/25]

N(t) = <-sint,-cost,0>

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