$\"\"$

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The vector function is $\"\"$ and $\"\"$ .

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Principal unit normal vector: Let $\"\"$ be the smooth curve represented by $\"\"$ on an open interval $\"\"$ , If $\"\"$ then principal unit normal vector is defined as, $\"\"$ .

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Where, $\"\"$ is derivative of unit tangent vector defined as $\"\"$ .

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$\"\"$

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Find the unit tangent vector $\"\"$ .

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Consider $\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Find the magnitude $\"\"$ :

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$\"\"$

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Substitute $\"\"$ and $\"\"$ values in $\"\"$

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$\"\"$

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Substitute $\"\"$ in above equation.

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$\"\"$

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$\"\"$

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Unit tangent vector $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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Find the magnitude $\"\"$ :

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$\"\"$

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Principal unit vector $\"\"$

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Substitute corresponding values in above equation.

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$\"\"$

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$\"\"$

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The vector function is $\"\"$ .

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$\"\"$ .

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$\"\"$ .

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Find $\"\"$ :

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$\"\"$

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At $\"\"$ ,

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$\"\"$ .

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$\"\"$

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Find $\"\"$ :

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$\"\"$

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At $\"\"$ ,

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$\"\"$

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$\"\"$

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$\"\"$ at $\"\"$ is $\"\"$ .

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$\"\"$ at $\"\"$ is $\"\"$ .

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$\"\"$ at $\"\"$ is $\"\"$ .

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$\"\"$ at $\"\"$ is $\"\"$ .

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