Find T, N, and B for the Given Space Curve $$\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 9 \right) \mathbf { i } + ( 2 \mathrm { t } - 9 ) \mathbf { j } + 4 \mathbf { k }$$

question 72

Multiple Choice

Find T, N, and B for the given space curve.
- $$\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 9 \right) \mathbf { i } + ( 2 \mathrm { t } - 9 ) \mathbf { j } + 4 \mathbf { k }$$

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

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Definitions:

Psychological Characteristics

Traits related to the mind and behavior that distinguish individuals from one another.

Phenotype

The observable characteristics or traits of an organism as a result of the interaction between its genetic makeup and the environment.

Genotypic Material

The genetic makeup of an organism, composed of DNA, that determines its characteristic features and properties.

Genotype

The genetic makeup of an individual organism, consisting of the alleles inherited from their parents.