Find the unit tangent vector of the given curve.
-$$r ( t ) = \left( 5 \sin ^ { 3 } 9 t \right) i + \left( 5 \cos ^ { 3 } 9 t \right) \mathbf { j }$$
A) T = (5 sin 9t)i - (5 cos 9t)j
B) T = (sin 9t)i - (cos 9t)j
C) T = (135 sin 9t)i -(135 cos 9t)j
D) T = (5 cos 9t)i - (5 sin 9t)j

Question

Quizplus · Accepted Answer

The unit tangent vector is found by differentiating the curve and normalizing the result. The derivative of $ r(t) $ is $ r'(t) = (135 \sin 9t \cos^2 9t)i - (135 \cos 9t \sin^2 9t)j $. Simplifying and normalizing gives $ T = (\sin 9t)i - (\cos 9t)j $.

Find the Unit Tangent Vector of the Given Curve r(t)=(5sin⁡39t)i+(5cos⁡39t)jr ( t ) = \left( 5 \sin ^ { 3 } 9 t \right) i + \left( 5 \cos ^ { 3 } 9 t \right) \mathbf { j }r(t)=(5sin39t)i+(5cos39t)j

Find the Unit Tangent Vector of the Given Curve $r ( t ) = \left( 5 \sin ^ { 3 } 9 t \right) i + \left( 5 \cos ^ { 3 } 9 t \right) \mathbf { j }$