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lect11_5

lect11_5 - Tangent and Normal Vector(11.5 1 Principal Unit...

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Tangent and Normal Vector - (11.5) 1. Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function r ! ! t " "# f ! t " , g ! t " , h ! t " \$ . The vector T ! ! t " " 1 r ! % ! t " r ! % ! t " is the unit tangent vector to the curve C . Define N ! ! t " " T ! % ! t " T % ! t " . The vector N ! ! t " is called the principal unit normal vector . Observe that ! N ! ! t " is a unit vector ! N ! ! t " is in the same direction as T ! % ! t " ! N ! ! t " and T ! ! t " are orthogonal It is known if || v ! ! t " || " c then v ! ! t " # v ! % ! t " " 0. Since T ! ! t " " 1, T ! ! t " # T ! % ! t " " 0. N ! ! t " # T ! ! t " " T ! % ! t " T % ! t " # T ! t " " 1 T % ! t " T ! % ! t " # T ! t " " 0 ! N ! ! t " " 1 ! dT ! ! t " ds N ! ! t " " T ! % ! t " T % ! t " " dT ! ! t " dt dT ! ! t " dt " dT ! ! t " ds ds dt dT ! ! t " ds ds dt " dT ! ! t " ds ds dt dT ! ! t " ds ds dt " dT ! ! t " ds dT ! ! t " ds " 1 ! dT ! ! t " ds , ds dt ! 0. N ! ! t " will always point to the direction in which T ! ! t " is turning as arc length increases and point to the concave side of the curve. Example Let the helix C be traced out by r " ! t " "# 2 cost , sint , t \$ . Find the unit tangent and principal unit normal vectors to the curve at anytime t . Sketch the helix C and the unit tangent and principal unit normal vectors when t " # 2 and t " # . r ! % ! t " " # " 2sin t , cos t , 1 \$ , r ! % ! t " " 4sin 2 t & cos 2 t & 1 T ! ! t " " 1 4sin 2 t & cos 2 t & 1 # " 2sin t , cos t , 1 \$" 1 3sin 2 t & 2 # " 2sin t , cos t , 1 \$ T ! % ! t " " 1 3sin 2 t & 2 # " 2cos t , " sin t , 0 \$ " 8sin t cos t " 2cos t sin t 2 3sin 2 t & 2 3/2 # " 2sin t , cos t , 1 \$ " 1 3sin 2 t & 2 # " 2cos t , " sin t , 0 \$ " 3sin t cos t 3sin 2 t & 2 3/2 # " 2sin t , cos t , 1 \$ 1

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x : " 2cos t 3sin 2 t & 2 & 6sin 2 t cos t " "
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lect11_5 - Tangent and Normal Vector(11.5 1 Principal Unit...

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