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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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Binormal VectorIn Exercises 9–11, use the binormal vector defined by the equation B = T × N .

Find the unit tangent, principal unit normal, and binormal vectors for the helix

r ( t ) = 4 cos t i + 4 sin t j + 3 t k

at t = π / 2 . Sketch the helix together with these three mutually orthogonal unit vectors.

To determine

To calculate: The unit tangent, principal unit normal, and binormal vectors for the given helix, r(t)=4costi+4sintj+3tk at π2.

Explanation

Given: The binormal vectors for the helix, r(t)=4costi+4sintj+3tk and the point t=π2.

Formula used: The unit normal vector is defined for, T(t)=r(t)r(t).

Calculation:

The helix equation is given below,

r(t)=4costi+4sintj+3tk

Differentiating the above equation,

r(t)=ddt(4costi+4sintj+3tk)=4(sint)i+4costj+3k=4sinti+4costj+3k

The magnitude of r(t) is,

r(t)=(4sint)2+(4cost)2+32=16((sint)2+(cost)2)+9=25=5

The unit normal vector is defined for,

T(t)=r(t)r(t)

Substituting all the value in the above equation,

T(t)=4sinti+4costj+3k5=45sinti+45costj+35k

At π2,

T(t)=45sin(π2)i+45cos(π2)j+35k=45i+35k

Differentiating the above equation,

T(t)=ddt(45sinti+45costj+35k)=45costi+45(sint)j+(0)k=45<

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