Find the normal vector N to r(t)=⟨t,cos t⟩ at t=(9 pi)/4. (Use symbolic notation and fractions where needed.)

cortejosni

cortejosni

Answered question

2022-08-10

Find the normal vector N to r ( t ) = t , cos t at t = 9 π 4 .
How do I find this normal vector? So basically I did what the feedback said. I found the derivative of each function in the vector and got.
( 1 , sin ( t ) )
Then I got the magnitude:
1 + sin 2 ( 9 π 4 )
Then I divided everything in my vector by that magnitude while putting in
9 π 4
like so:
sin ( 9 π 4 ) 1 + sin 2 ( 9 π 4 )
What my problem? Here's my answer:
2 3 , 3 3

Answer & Explanation

Paulina Horne

Paulina Horne

Beginner2022-08-11Added 10 answers

The unit tangent vector is
T ( t ) = ( 1 1 + sin 2 ( t ) , sin ( t ) sin 2 ( t ) + 1 )
So the normal vector of r(t) is
T ( t ) = ( 2 sin ( 2 t ) ( 3 cos ( 2 t ) ) 3 2 , cos ( t ) ( 1 + sin 2 ( t ) ) 3 2 )
Now let t = 9 π 4 and you should get
T ( 9 π 4 ) = ( 6 9 , 2 3 9 ) .

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