Find T, N, and B for the Given Space Curve $$r ( t ) = \left( 8 + 6 \sin \frac { 2 } { 3 } t \right) i + \left( 4 + 6 \cos \frac { 2 } { 3 } t \right) j + 3 t k$$

question 55

Multiple Choice

Find T, N, and B for the given space curve.
- $$r ( t ) = \left( 8 + 6 \sin \frac { 2 } { 3 } t \right) i + \left( 4 + 6 \cos \frac { 2 } { 3 } t \right) j + 3 t k$$

• A) \(\mathrm { T } = \frac { 4 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { i } - \frac { 4 } { 5 } ( \sin 0.667 \mathrm { t } ) \mathbf { j } ; \mathbf { N } = ( - \sin 0.667 \mathrm { t } ) \mathbf { i } - ( \cos 0.667 \mathrm { t } ) \mathbf { j } ; \mathbf { B } = \frac { 3 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { i } - \frac { 3 } { 5 } ( \sin 0.667 \mathrm { t } ) \mathbf { j } -\) \(\frac { 4 } { 5 } \mathbf { k }\)
• B) \(\mathrm { T } = \frac { 4 } { 5 } ( \sin 0.667 \mathrm { t } ) \mathbf { i } - \frac { 4 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { j } ; \mathbf { N } = ( - \sin 0.667 \mathrm { t } ) \mathbf { i } - ( \cos 0.667 \mathrm { t } ) \mathbf { j } ; \mathbf { B } = \frac { 3 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { i } - \frac { 3 } { 5 } ( \sin 0.667 \mathrm { t } ) \mathbf { j } -\) \(\frac { 4 } { 5 } \mathbf { k }\)
• C) \(\mathrm { T } = \frac { 4 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { i } - \frac { 4 } { 5 } ( \sin 0.667 \mathrm { t } ) \mathbf { j } + \frac { 3 } { 5 } \mathbf { k } ; \mathbf { N } = ( - \sin 0.667 \mathrm { t } ) \mathbf { i } - ( \cos 0.667 \mathrm { t } ) \mathbf { j } ; \mathbf { B } = - \frac { 4 } { 5 } \mathbf { k }\)
• D) \(\mathrm { T } = \frac { 4 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { i } - \frac { 4 } { 5 } ( \sin 0.667 \mathrm { t } ) \mathbf { j } + \frac { 3 } { 5 } \mathbf { k } ; \mathbf { N } = ( - \sin 0.667 \mathrm { t } ) \mathbf { i } - ( \cos 0.667 \mathrm { t } ) \mathbf { j } ; \mathbf { B } = \frac { 3 } { 5 } ( \cos 0.667 \mathrm { t } ) \mathbf { i } - \frac { 3 } { 5 } ( \sin\) \(0.667 t ) \mathbf { j } - \frac { 4 } { 5 } \mathbf { k }\)

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