91Ó°ÊÓ

In physics, the displacement vector characterizes the change in position of an object. It provides both the direction and the magnitude of the movement between two points. This is different from distance, which only describes how much ground an object has covered, regardless of its direction.

To find the vector displacement, you subtract the initial position vector from the final position vector. Consider a scenario where an object moves from an initial position \( \vec{r}_1 \) to a final position \( \vec{r}_2 \). The displacement \( \vec{d} \) is given by the formula:
\[ \vec{d} = \vec{r}_2 - \vec{r}_1 \]

It's important to remember:

• Direction Matters: Displacement considers the direction of the movement, making it a vector quantity.
• Initial and Final Positions: Ensure correct vector subtraction by aligning all vector components (i.e., \( \, \hat{\mathrm{i}} \, , \, \hat{\mathrm{j}} \, , \, \hat{\mathrm{k}} \, \)).
• Units: Displacement is measured in meters (m).

In our case, the displacement vector is calculated as \( \vec{d} = (-6.80 \, \hat{\mathrm{i}} + 6.20 \, \hat{\mathrm{j}} - 0.10 \, \hat{\mathrm{k}}) \, \text{m} \). This indicates that the object moved predominantly in the negative x-direction (\( \hat{\mathrm{i}} \)), positively in the y-direction (\( \hat{\mathrm{j}} \)), and slightly moved down in the z-direction (\( \hat{\mathrm{k}} \)).

The dot product is a fundamental mathematical operation used to calculate the amount of one vector that extends in the direction of another. It's particularly useful in physics to determine the work done by a force acting on an object during its displacement. The dot product is calculated by multiplying the corresponding components of two vectors, and summing these products.

The typical formula for finding the dot product \( \vec{A} \cdot \vec{B} \) is:
\[ \vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z \]

The dot product can define the work \( W \) done by a force \( \vec{F} \) along a certain displacement \( \vec{d} \):
\[ W = \vec{F} \cdot \vec{d} \]

Key aspects of the dot product include:

• No Direction Content: The dot product is a scalar. It provides magnitude but no directional information.
• Non-Perpendicular Contribution: Only the portion of vectors that are in the same direction contributes to the dot product.

In this problem, \( W = 41.67 \, \text{J} \), which reflects the amount of work performed by the force on the object.

Average power is a representation of the work done by a force over time. It's an important concept in understanding how energy is transferred per unit of time. By determining average power, we understand not only the work done but also the speed of this energy conversion.

The formula for average power \( P_{\text{avg}} \) is:
\[ P_{\text{avg}} = \frac{W}{\Delta t} \]

Where:

• \( W \) represents the work done, measured in Joules (J).
• \( \Delta t \) is the time interval during which the work was done, measured in seconds (s).

This formula emphasizes:

• Time Dependency: Power depends on both the work done and the time taken; greater power means faster energy transfer.
• Units: Power is measured in watts (W), with 1 W equating to 1 Joule per second.

Following through on our example, we calculate the average power to be around \( 19.84 \) watts, signifying that energy was transferred at a rate of roughly 19.84 Joules per second over the given time interval.