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

question 1

Multiple Choice

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ .
$\mathbf { r } ( t ) = ( 7 + 3 t ) \mathbf { i } + ( 10 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

• A) \(\mathbf { r } ( t ( s ) ) = \left( 7 - \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }\)
• B) \(\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 - \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }\)
• C) \(\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - \left( \frac { 6 s } { \sqrt { 126 } } \right) \mathbf { k }\)
• D)
• \(\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } + ( 6 s ) \mathbf { k }\)
• E)
• \(\mathbf { r } ( t ( s ) ) = \left( 7 + \frac { 3 } { \sqrt { 126 } } s \right) \mathbf { i } + \left( 10 + \frac { 9 } { \sqrt { 126 } } s \right) \mathbf { j } - ( 6 s ) \mathbf { k }\)

Understand the theories of intelligence, including multiple intelligences and the factors that influence intellectual development in children.

Understand the complexity and dynamics of relationships in middle age, including the transition stages and effects on identity and relationships.

Comprehend the challenges and benefits of marital relationships over time, including marital satisfaction and the impact of life stages on marriage.

Recognize the experiences and challenges faced by homosexual individuals in middle age, particularly in terms of coming out and relationship dynamics.

---

Definitions:

Reciprocal Behavior

Mutual action or exchange between parties, where each party responds to the other with a similar kind of behavior.

Reciprocally Related

Being mutually influencing or having a two-way relationship where both parties have an effect on each other.

Attribution Theory

A psychological theory that explains how individuals pinpoint the causes of their own and others' behavior.

Internal

Relating to or occurring within an organization or entity.