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How do you determine the unit normal vector?

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Quick Answer

To find the unit normal vector, first find the unit tangent vector. Finding the unit tangent vector requires differentiating each component of a vector function, as well as the length of the vector function. Finally, use the derivative of the unit tangent vector over the length of the unit tangent vector to determine the unit normal vector.

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Full Answer

  1. Differentiate the vector function, and set it to zero

    First, differentiate the given vector function: r(t) = cos ti + sin tj + tk. The derivative of this function is r'(t) = -sin ti + cos tj + k. The vector function is the point where t = 0 is: r'(0) = j + k.

  2. Find the length of the vector function

    The length of the vector function is found by determining the square root of the square of each directional coefficient in the derivative: |r'(t)| = sqrt(1^2 + 1^2) = sqrt(2). Putting all of these elements together, the unit tangent vector is: T(t) = r'(t) / |r'(t)| = (1/sqrt(2)) * (-sin ti + cos tj + k).

  3. Differentiate the unit tangent vector, and set it to zero

    Find the derivative of the unit tangent vector: T'(t) = (1/sqrt(2)) * (-cos ti - sin tj). Setting this to zero gives the result: T'(0) = -1/sqrt(2).

  4. Find the length of the unit tangent vector

    For the unit tangent vector, the length is: |T'(t)| = (1/sqrt(2)).

  5. Find the unit normal vector

    The unit normal vector is the unit tangent vector over the length of the unit tangent vector: N(t) = T'(t)/|T'(t)| = -cos ti - sin tj.

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