   Chapter 13.3, Problem 48E

Chapter
Section
Textbook Problem

# Find the vectors T, N, and B at the given point.48. r(t) = ⟨cos t, sin t, ln cos t⟩, (1, 0, 0)

To determine

To find: The vectors T, N, and B of vector function r(t)=cost,sint,lncost at point (1,0,0) .

Explanation

Given data:

Vector function r(t)=cost,sint,lncost and point (1,0,0) .

Formula used:

Write the expression for tangent vector of a vector function r(t) (T(t)) .

T(t)=r(t)|r(t)| (1)

Here,

r(t) is first derivative of function r(t) .

Write the expression for normal vector of vector function r(t) (N(t)) .

N(t)=T(t)|T(t)| (2)

Here,

T(t) is first derivative of tangent function.

Write the expression for binormal vector of vector function r(t) (B(t)) .

B(t)=T(t)×N(t) (3)

Consider the two three-dimensional vector functions such as u(t)=u1(t),u2(t),u3(t) and v(t)=v1(t),v2(t),v3(t) .

Cross product of vectors:

Write the expression for cross product of vectors u(t) and v(t) (u(t)×v(t)) .

u(t)×v(t)=|ijku1(t)u2(t)u3(t)v1(t)v2(t)v3(t)|=[(u2(t)v3(t)v2(t)u3(t))],[(u1(t)v3(t)v1(t)u3(t))],[(u1(t)v2(t)v1(t)u2(t))]

Write the expression for magnitude of vector a (|a|) .

|a|=a12+a22+a32

Here,

a1 , a2 and a3 are the x, y, and z-coordinates of vector respectively.

Write the vector function.

r(t)=cost,sint,lncost (4)

Equate the components of r(t) with point (1,0,0) .

cost=1t=cos1(1)t=0

sint=0t=sin1(0)t=0

Hence, the value of t is 0.

Apply differentiation with respect to t on both sides of equation (4).

r(t)=ddtcost,sint,lncost=ddt(cost),ddt(sint),ddt(lncost)=sint,cost,1cost(sint){ddx(cosx)=sinx,ddx(lnx)=1x,ddx(sinx)=cosx}=sint,cost,tant

Find the value of |r(t)| .

|r(t)|=(sint)2+(cost)2+(tant)2=sin2t+cos2t+tan2t=1+tan2t{sin2x+cos2x=1}=sec2t{1+tan2x=sec2x}

|r(t)|=sect

Substitute sint,cost,tant for r(t) and sect for |r(t)| in equation (1),

T(t)=sint,cost,tantsect=sintsect,costsect,tantsect=sint(1cost),cost(1cost),(sintcost)(1cost){secx=1cosx}

T(t)=sintcost,cos2t,sint (5)

Substitute 0 for t,

T(0)=sin(0)cos(0),cos2(0),sin(0)=(0)(1),12,0=0,1,0

Apply differentiation with respect to t on both sides of equation (5)

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