Find T, N, and B for the Given Space Curve $$r ( t ) = \frac { 4 } { 3 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 4 } { 3 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + 1 \mathrm { tk }$$

question 97

Multiple Choice

Find T, N, and B for the given space curve.
- $$r ( t ) = \frac { 4 } { 3 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 4 } { 3 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + 1 \mathrm { tk }$$

• A) \(T = \frac { 2 } { 3 } \sqrt { 1 + t } \mathbf { i } + \frac { 2 } { 3 } \sqrt { 1 - t } j + \frac { 1 } { 3 } k ; N = \frac { 1 } { 2 } \sqrt { 2 - 2 } \mathrm { ti } + \frac { 1 } { 2 } \sqrt { 2 + 2 t j } ; B = \frac { 1 } { 6 } \sqrt { 2 + 2 t i } + \frac { 1 } { 6 } \sqrt { 2 - 2 } \mathbf { j } + \frac { 2 \sqrt { 2 } } { 3 } \mathbf { k }\)
• B) \(\mathrm { T } = \frac { 2 } { 3 } \sqrt { 1 + \mathrm { t } } \mathrm { i } - \frac { 2 } { 3 } \sqrt { 1 - \mathrm { t } } \mathrm { j } + \frac { 1 } { 3 } \mathrm { k } ; \mathrm { N } = \frac { 1 } { 2 } \sqrt { 2 - 2 \mathrm { t } } + \frac { 1 } { 2 } \sqrt { 2 + 2 \mathrm { t } } ; \mathbf { B } = - \frac { 1 } { 6 } \sqrt { 2 + 2 \mathrm { t } } \mathrm { i } + \frac { 1 } { 6 } \sqrt { 2 - 2 \mathrm { t } } \mathbf { j } + \frac { 2 \sqrt { 2 } } { 3 } \mathrm { k }\)
• C) \(\mathrm { T } = 2 \sqrt { 1 + \mathrm { t } } - 2 \sqrt { 1 - t } \mathbf { j } + 1 \mathbf { k } ; \mathbf { N } = \frac { 3 } { 2 } \sqrt { 2 - 2 \mathrm { t } } + \frac { 3 } { 2 } \sqrt { 2 + 2 t } \mathbf { j } ; \mathbf { B } = - \frac { 1 } { 2 } \sqrt { 2 + 2 t \mathrm { i } } + \frac { 1 } { 2 } \sqrt { 2 - 2 \mathrm { t } } + 2 \sqrt { 2 } \mathbf { k }\)
• D) \(\mathrm { T } = \frac { 1 } { 3 } \sqrt { 1 + \mathrm { t } } - \frac { 1 } { 3 } \sqrt { 1 - \mathrm { t } } \mathrm { j } + \frac { 1 } { 3 } \mathrm { k } ; \mathrm { N } = \frac { 1 } { 2 } \sqrt { 2 - 2 \mathrm { t } } \mathrm { i } + \frac { 1 } { 2 } \sqrt { 2 + 2 \mathrm { t } } ; \mathbf { B } = - \frac { 1 } { 6 } \sqrt { 2 + 2 \mathrm { t } } \mathrm { i } + \frac { 1 } { 6 } \sqrt { 2 - 2 \mathrm { t } } + \frac { \sqrt { 2 } } { 3 } \mathbf { k }\)

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