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A position vector is a crucial concept in physics and mathematics that helps describe the location of a particle in space relative to a fixed point, often referred to as the origin. In this context, the position vector expresses the exact position of a moving particle at any given time. It's represented in a three-dimensional space using components aligned along the axes denoted by unit vectors

• \( \hat{\mathrm{i}} \): along the x-axis
• \( \hat{\mathrm{j}} \): along the y-axis
• \( \hat{\mathrm{k}} \): along the z-axis

The given position vector \( \overrightarrow{\mathrm{r}} = 3 \hat{\mathrm{i}} + 4t^2 \hat{\mathrm{j}} - t^3 \hat{\mathrm{k}} \) indicates the particle's placement with respect to time \( t \). Each component reflects how the position changes in its specific direction, offering a concise mathematical representation of physical space.

Understanding the components of vectors is essential when analyzing vector quantities. Vectors can have different magnitudes in each direction, which are its components. In a three-dimensional space, vectors have components along the x, y, and z axes. The components serve as building blocks to determine the resultant effect of the vector.
Let's break down the given position vector:

• The \( i \)-component is constant and remains \( 3 \hat{\mathrm{i}} \), meaning there's no change along the x-axis over time.
• The \( j \)-component, \( 4t^2 \hat{\mathrm{j}} \), increases with the square of time, indicating acceleration in the y-direction.
• The \( k \)-component, \( -t^3 \hat{\mathrm{k}} \), decreases with the cube of time, exhibiting an increasing negative movement along the z-axis.

Components make it easier to perform calculations, such as addition or subtraction of vectors, by handling each part independently. This principle is pivotal when calculating the resultant displacement from initial and final position vectors.

Displacement is a vector that represents the change in position of an object. It is independent of the path taken and depends solely on the initial and final positions. In vector terms, displacement \( \overrightarrow{\mathrm{d}} \) can be calculated as the difference between the final position vector \( \overrightarrow{\mathrm{r}}(t_2) \) and the initial position vector \( \overrightarrow{\mathrm{r}}(t_1) \).
This case involves a time interval from \( t=1 \mathrm{~s} \) to \( t=3 \mathrm{~s} \).
Substituting time values:

• Initial position: \( \overrightarrow{\mathrm{r}}(1) = 3\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - \hat{\mathrm{k}} \)
• Final position: \( \overrightarrow{\mathrm{r}}(3) = 3\hat{\mathrm{i}} + 36\hat{\mathrm{j}} - 27\hat{\mathrm{k}} \)

Displacement vector calculation:
\[\overrightarrow{\mathrm{d}} = \overrightarrow{\mathrm{r}}(3) - \overrightarrow{\mathrm{r}}(1) = (3\hat{\mathrm{i}} + 36\hat{\mathrm{j}} - 27\hat{\mathrm{k}}) - (3\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - \hat{\mathrm{k}})\]
Simplified, the displacement is \( 0\hat{\mathrm{i}} + 32\hat{\mathrm{j}} - 26\hat{\mathrm{k}} \). This indicates the shift in the particle's position over the specified time interval.

In physics, a time interval is the difference between two points in time. It is important when studying motion, as changes in quantities like velocity or position often depend on the time elapsed.
For this problem, the time interval analyzed is from \( t=1 \) second to \( t=3 \) seconds. Understanding time intervals allows physicists to calculate changes and predict future states of objects in motion.
Key aspects of time intervals in motion calculations:

• Time intervals help determine how much an object's position or velocity has changed.
• In combination with position vectors, they are used to find displacement, which highlights how far and in what direction an object has moved.
• They also play a role in calculating average velocities and accelerations, aiding in comprehending an object's dynamics more thoroughly.

In this exercise, analyzing movement over the \( 2 \)-second interval allowed for precise calculation of displacement, reflecting the true change in position.