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Chapter 11: Q. 30 (page 880)

Find the unit tangent vector and the principal unit normal vector at the specified value of t.

r(t)=(asinht,bcosht)Whereaandbarepositive,t=0

Short Answer

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Step by step solution

01

Step1. Given Information

n xm m x

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Most popular questions from this chapter

Statewhatitmeansforavectorfunctionr(t)=x(t),y(t),z(t)tobeintegrable.

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=cos3t,sin3t,t=蟺4

As we saw in Example 1, the graph of the vector-valued function r(t)=cost,sint,t,fort鈭�[0,2蟺] is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

Evaluate the limits in Exercises 42鈥�45.

limt鈫�0+(sintt,1-costt,(1+1t)t)

Let r(t)=x(t),y(t),z(t)be a differentiable vector function on some interval I鈯�Rsuch that the derivative of the unit tangent vector T(t0)=0, where t0鈭�I. Prove that the binormal vectorB(t0)=T(t0)脳N(t0)

(a) B(t0)is a unit vector;

(b)B(t0)is orthogonal to both T(t0)and N(t0).

Also, prove that T(t0),N(t0), and B(t0)form a right-handed coordinate system.
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