Reparametrize the Curve with Respect to Arc Length Measured from the Point

question 12

Multiple Choice

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ .
$\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

• A) \(\mathbf { r } ( t ( s ) ) = \left( 5 - \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }\)
• \(\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 - \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }\)
• B)
• \(\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } + ( 6 s ) \mathbf { k }\)
• C)
• \(\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }\)
• D)
• \(\mathbf { r } ( t ( s ) ) = \left( 5 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 8 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - ( 6 s ) \mathbf { k }\)

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Definitions:

Ethical Dilemma

A situation in which a difficult choice has to be made between two or more actions, where none of the options resolve the ethical principles or standards at stake fully.

Kohlberg's Moral Development Theory

A theory that proposes individuals progress through different stages of moral reasoning, from basic obedience to societal rules to higher-level ethical principles.

Conventional Level

A stage in moral development proposed by Lawrence Kohlberg, where an individual's sense of morality is tied to personal and societal relationships, and compliance with laws and norms.

Terminal Values

Fundamental goals or end-states that individuals strive to achieve in their lives, guiding their actions and judgments.