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# Unit Normal Vector

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Let vector r(t) = t i + t^2 j represent a plane curve.
Find T(t), T(1) and N(1). Sketch the plane curve and graph the vectors T(1) and N(1) at the point t = 1.

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https://brainmass.com/math/vector-calculus/unit-normal-vector-planes-70321

#### Solution Preview

Please see the attached file.

SOLUTION

To determine unit normal vector for the curve r(t) = t i + t2 j ....(1)

Step 1 : Determine the tangent vector by differentiating (1)

dr/dt = T(t) = i + 2tj ...........(2)

Step 2 : Determine unit tangent vector by dividing right hand side of (2) by the magnitude of the vector
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Å¤(t)= (i + 2tj)/âˆš1 + 4t2 ..........(3)

Step 3 : Determine normal vector by differentiating (3)

N(t) = dÅ¤(t)/dt

To differentiate (3) we use the following rule :

d/dx[f(x)/g(x)] = [g(x)df(x)/dx - f(x)dg(x)/dx]}/g(x)2

= (Denominator x derivative of numerator - Numerator x derivative of denominator)/Denominator2

Derivative of numerator = d/dt (i + 2tj) = 2j ......(4)
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Derivative of denominator = d/dt âˆš1 + ...

#### Solution Summary

The vector functions and plane curve representations are determined. The expert sketches the plane curve and graphs the vectors.

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