1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
BARSIC [14]
3 years ago
14

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

Mathematics
1 answer:
const2013 [10]3 years ago
6 0

Answer:

a.

T(t) = ( -sin(t^2), cos(t^2) )\\\\N(t) = T'(t) / |T'(t) |  =   (-cos(t^2) , -sin(t^2))

b.

T(t) =r'(t)/|r'(t)| =  (2t/ \sqrt{4t^2   + 36} , -6/\sqrt{4t^2   + 36},0)

N(t) =T(t)/|T'(t)| =  (3/(9 + t^2)^{1/2} , t/(9 + t^2)^{1/2},0)

Step-by-step explanation:

Remember that for any curve      r(t)  

The tangent vector is given by

T(t) = \frac{r'(t) }{| r'(t)| }

And the normal vector is given by

N(t) = \frac{T'(t)}{|T'(t)|}

a.

For this case, using the chain rule

r'(t) = (  -10*2tsin(t^2) ,   102t cos(t^2)   )\\

And also remember that

|r'(t)| = \sqrt{(-10*2tsin(t^2))^2  +  ( 10*2t cos(t^2) )^2} \\\\       = \sqrt{400 t^2*(  sin(t^2)^2  +  cos(t^2) ^2 })\\=\sqrt{400t^2} = 20t

Therefore

T(t) = r'(t) / |r'(t) | =  (  -10*2tsin(t^2) ,   10*2t cos(t^2)   )/ 20t\\\\ = (  -10*2tsin(t^2)/ 20t ,   10*2t cos(t^2) / 20t  )\\= ( -sin(t^2), cos(t^2) )

Similarly, using the quotient rule and the chain rule

T'(t) = ( -2t cos(t^2) , -2t sin(t^2))

And also

|T'(t)| = \sqrt{  ( -2t cos(t^2))^2 + (-2t sin(t^2))^2} = \sqrt{ 4t^2 ( ( cos(t^2))^2 + ( sin(t^2))^2)} = \sqrt{4t^2} \\ = 2t

Therefore

N(t) = T'(t) / |T'(t) |  =   (-cos(t^2) , -sin(t^2))

Notice that

1.   |N(t)| = |T(t) | = \sqrt{ cos(t^2)^2  + sin(t^2)^2 } = \sqrt{1} =  1

2.   N(t)*T(T) = cos(t^2) sin(t^2 ) - cos(t^2) sin(t^2 ) = 0

b.

Simlarly

r'(t) = (2t,-6,0) \\

and

|r'(t)| = \sqrt{(2t)^2   + 6^2} = \sqrt{4t^2   + 36}

Therefore

T(t) =r'(t)/|r'(t)| =  (2t/ \sqrt{4t^2   + 36} , -6/\sqrt{4t^2   + 36},0)

Then

T'(t) = (9/(9 + t^2)^{3/2} , (3 t)/(9 + t^2)^{3/2},0)

and also

|T'(t)| = \sqrt{ ( (9/(9 + t^2)^{3/2} )^2 +   ( (3 t)/(9 + t^2)^{3/2})^2  +  0^2 }\\= 3/(t^2 + 9 )

And since

N(t) =T(t)/|T'(t)| =  (3/(9 + t^2)^{1/2} , t/(9 + t^2)^{1/2},0)

You might be interested in
6 The radius of a circle is 36 millimeters.
love history [14]

Answer:

G. 226 mm

Step-by-step explanation:

C=2(3.14)(36)

8 0
3 years ago
Read 2 more answers
The blades of a windmill turn on an axis that is 35 feet above the ground. The blades are 10 feet long and complete two rotation
7nadin3 [17]
<span>It could be C. If the blade spends twice in a minute, it would be 30 seconds. 30=2pi/B=pi/15. </span>
6 0
3 years ago
Read 2 more answers
(Please help HURRY!) (Don't delete this!) (I WILL GIVE BRAINY!)
ipn [44]
1. A or B

2. C

3. I think B
4 0
3 years ago
Read 2 more answers
7) 2x + y = 7<br> What is the intercept
posledela

Answer:

2x + y = 7

x=7/2

y=7

Step-by-step explanation:

to find y itercept =0

to find x intercept=0

5 0
3 years ago
Write the ordered pair for a point that is 9 units to the left of the origin and lies on the x-axis.
natulia [17]
(((((((((-9,0)))))))))
8 0
3 years ago
Other questions:
  • How do you solve x^2+4x-5=0
    14·2 answers
  • How many different ways can a family visit Philadelphia, Baltimore, and Washington? 3 6 12
    9·2 answers
  • If DH = 4x – 10 and FH = x + 8, find the value of x for which DEFG must be a parallelogram.
    11·2 answers
  • Carmen is renting buses for a school field trip. The number of buses needed depends on how many students are going on the trip.
    7·1 answer
  • Someone please help me with my last question
    11·2 answers
  • Prime factorization of 52 using factor tree
    10·1 answer
  • Solve the system by substitution. y=2x y=10x-16
    6·2 answers
  • Write a division problem with 1/4 as the dividend and 3 as the divisor. Then, find the quotient.
    13·2 answers
  • Can someone please help me
    8·1 answer
  • Half of the children in a library group like pets, one fourth don't like pets and seven children do not have any
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!