$\"\"$

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The frenet serret formulas are $\"\"$ , $\"\"$ and $\"\"$ .

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(a)

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Prove that $\"\"$ .

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Consider $\"\"$ .

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$\"\"$ .

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Since $\"\"$

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$\"\"$ .

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Apply derivative on each side.

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$\"\"$

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Apply the formula : $\"\"$ .

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$\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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Consider $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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(b)

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Prove that $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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The cross product of a vector itself is zero.

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(c)

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Prove that $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side.

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$\"\"$

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Apply formula : $\"\"$ . and $\"\"$ .

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$\"\"$

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$\"\"$

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Consider $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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(d)

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Prove that $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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Since $\"\"$ .

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$\"\"$

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$\"\"$

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Consider $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ \ \

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$\"\"$ . $\"\"$

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(a) $\"\"$ .

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(b) $\"\"$ .

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(c) $\"\"$ .

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(d) $\"\"$ .