L> The Unit Tangent and also the Unit typical Vectors
The Unit Tangent and also the Unit common VectorsThe Unit Tangent VectorThe derivative that a vector valued role gives a new vector valued functionthat is tangent to the identified curve. The analogue to the steep ofthe tangent line is the direction that the tangent line. Due to the fact that a vectorcontains a magnitude and also a direction, the velocity vector has moreinformation 보다 we need. We can strip a vector the its magnitude bydividing by its magnitude. meaning of the Unit Tangent VectorLet r(t) be a differentiable vector valuedfunction and v(t) = r"(t) it is in the velocity vector. Thenwe define the unit tangent vector by together the unit vector in the direction the the velocity vector.v(t)T(t) = ||v(t)||
ExampleLet r(t)= t i + et j - 3t2 kFindthe T(t) and also T(0).

You are watching: Find t(t), n(t), at, and an at the given time t for the curve r(t). r(t) = t2i + 2tj, t = 1

SolutionWehavev(t) = r"(t) = i + et j- 6t kand
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To find the unit tangent vector, we just divide
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Tofind T(0) plugin 0 toget
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The principal Unit normal Vector

A normal vector is a perpendicular vector. Offered a vector v in thespace, there room infinitely numerous perpendicular vectors. Our score is toselect a special vector that is normal to the unit tangent vector.Geometrically, for a non directly curve, this vector is the distinct vector thatpoint right into the curve. Algebraically we have the right to compute the vector making use of thefollowing definition.
Definition that the principal Unit typical Vector

Let r(t) it is in a differentiable vector valued role and let T(t) it is in the unit tangent vector. Climate the major unit regular vector N(t) is identified by

T"(t) N(t) = ||T"(t)||
Comparing this through the formula because that the unittangent vector, if we think of the unit tangent vector together a vector valuedfunction, then the primary unit typical vector is the unit tangent vector ofthe unit tangent vector function. You will find that finding the principalunit common vector is nearly always cumbersome. The quotient dominance usuallyrears the ugly head.ExampleFindthe unit regular vector for the vector valued functionr(t) = ti + t2 jandsketch the curve, the unit tangent and also unit normal vectors once t= 1.SolutionFirstwe discover the unit tangent vector
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Nowuse the quotient ascendancy to uncover T"(t)
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Sincethe unit vector in the direction of a given vector will be the same aftermultiplying the vector by a hopeful scalar, we have the right to simplify by multiplying bythe factor
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Thefirst element gets rid of the denominator and the 2nd factor gets rid the thefractional power. Us have
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Nowwe divide by the size (after first dividing through 2)to get
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Nowplug in 1 because that both the unit tangent vector to get
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Thepicture listed below shows the graph and the two vectors.

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Tangential and Normal contents ofAccelerationImagine yourself driving under from EchoSummit in the direction of Myers and having your brakes fail. Together you room riding youwill endure two pressures (other than the force of terror) that will adjust thevelocity. The pressure of gravity will reason the car to rise inspeed. A second change in velocity will certainly be led to by the auto going aroundthe curve. The an initial component the acceleration is called the tangentialcomponent the acceleration and the secondis called the typical component of acceleration.As you might guess the tangential component of acceleration is in the direction ofthe unit tangent vector and the normal component the acceleration is in thedirection of the primary unit regular vector. As soon as we have T and also N,it is straightforward to find the two components. We have

Tangential and also Normal materials of Acceleration

The tangential ingredient of acceleration is

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and the regular component of acceleration is

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and

a = aNN + aTT

ProofFirst noticethat v= ||v|| Tand T"= ||T"|| NTaking the derivative of bothsides givesa = v" = ||v||" T + ||v|| T"= ||v||" T + ||v|| ||T" || NThistells us that the acceleration vector is in the airplane that contains the unittangent vector and also the unit normal vector. The very first equalityfollows instantly from the an interpretation of the componentof a vector in the direction of one more vector. The 2nd equalitieswill it is in left as exercises.ExampleFindthe tangential and normal contents of acceleration because that the prior exampler(t) = ti + t2 jSolutionTakingtwo derivatives, we havea(t) = r""(t) = 2jWedot the acceleration vector with the unit tangent and also normal vectors come get
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