Find T, N, and B for the Given Space Curve

question 79

Multiple Choice

Find T, N, and B for the given space curve.
-r(t) = (cosh t) i + (sinh t) j + tk

• A) \(\mathbf { T } = - \frac { \sqrt { 2 } } { 2 } ( \tanh \mathrm { t } \operatorname { sech } \mathrm { t } ) \mathbf { i } + \frac { \sqrt { 2 } } { 2 } \mathbf { j } + \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } \mathrm { t } ) \mathbf { k } ; \mathbf { N } = ( - \operatorname { sech } \mathrm { t } ) \mathbf { i } - ( \tanh \mathrm { t } ) \mathbf { k }\); \(\mathbf { B } = \frac { \sqrt { 2 } } { 2 } ( \sinh \mathrm { t } ) \mathbf { i } - \frac { \sqrt { 2 } } { 2 } \mathrm { j } - \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } t ) \mathbf { k }\)
• B) \(\mathbf { T } = \frac { \sqrt { 2 } } { 2 } ( \tanh t \operatorname { sech } t ) \mathbf { i } + \frac { \sqrt { 2 } } { 2 } \mathbf { j } + \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } t ) \mathbf { k } ; \mathbf { N } = \left( \operatorname { sech } ^ { 2 } t \right) \mathbf { i } - ( \sinh t ) \mathbf { k } ;\) \(\mathbf { B } = \frac { \sqrt { 2 } } { 2 } ( \sinh t ) \mathbf { i } + \frac { \sqrt { 2 } } { 2 } \mathbf { j } + \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } t ) \mathbf { k }\)
• C) \(\mathbf { T } = - \frac { \sqrt { 2 } } { 2 } ( \tanh t ) i + \frac { \sqrt { 2 } } { 2 } j + \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } t ) \mathbf { k } ; \mathbf { N } = ( - \operatorname { sech } t ) i - ( \sinh t ) \mathbf { k }\); \(\mathbf { B } = - \frac { \sqrt { 2 } } { 2 } ( \sinh t ) \mathbf { i } + \frac { \sqrt { 2 } } { 2 } \mathbf { j } - \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } t ) \mathbf { k }\)
• D) \(\mathbf { T } = \frac { \sqrt { 2 } } { 2 } ( \tanh \mathrm { t } ) \mathbf { i } + \frac { \sqrt { 2 } } { 2 } \mathbf { j } + \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } t ) \mathbf { k } ; \mathbf { N } = ( \operatorname { sech } t ) \mathbf { i } - ( \tanh t ) \mathbf { k } ; \mathbf { B } = - \frac { \sqrt { 2 } } { 2 } ( \tanh \mathrm { t } ) \mathbf { i } + \frac { \sqrt { 2 } } { 2 } \mathbf { j } - \frac { \sqrt { 2 } } { 2 } ( \operatorname { sech } \mathrm { t } ) \mathbf { k }\)

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