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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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    Solution Preview

    Please see the attached file.


    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
    Ť(t)= (i + 2tj)/√1 + 4t2 ..........(3)

    Step 3 : Determine normal vector by differentiating (3)

    N(t) = dŤ(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)
    Derivative of denominator = d/dt √1 + ...

    Solution Summary

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