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Vector displacement is a fundamental concept in physics that describes the change in position of an object. In our problem, we need to determine how the package moved from its starting point to its finishing point. This is done by calculating the displacement vector, which is essentially the difference between the initial and final positions.
To find vector displacement, we take the vector representing the final position and subtract the vector representing the initial position. This operation is straightforward, requiring the subtraction of corresponding components:

• Subtract the initial x-coordinate from the final x-coordinate to find the i-component.
• Do the same for the y- and z-components to find the j- and k-components, respectively.

The resulting vector tells us not just the distance but the overall movement direction of the package in three-dimensional space. This calculation is key in many physics problems, as it helps us understand the path an object takes over time.

The dot product is a crucial operation when working with vectors in physics, particularly when calculating work done by a force across a distance. Essentially, it allows us to determine how much of one vector goes in the direction of another. In our exercise, we use the dot product to calculate the work done by the machine's force on the package.
To compute the dot product, we multiply corresponding components of two vectors you want to analyze:

• Multiply the i-components of the force and the displacement vectors.
• Do the same for the j- and k-components.
• Add all these products together.

The resultant scalar (a single number) from these calculations reflects the amount of energy transferred by the force along the direction of the displacement. Understanding the dot product helps explain how different vectors relate to each other in practical scenarios, such as applying a force to move an object.

Average power is another important concept, representing the rate at which work is done or energy is transferred over a given time period. In this problem, we need to determine how quickly the machine's force did the work of moving the package.
The formula for average power is quite simple: it's the total work done divided by the time taken to do this work:

• Identify the total amount of work completed, which we calculated using the dot product of force and displacement vectors.
• Divide this work by how long it took to accomplish (in seconds, minutes, etc.).

This calculation tells us about the efficiency and performance of the system. A significant concept in many real-world applications, average power provides insight into how systems operate under various forces and movements.

Tackling physics problems involves several steps, using both math and critical thinking. Begin by understanding the problem and identifying what is asked. Each physics problem, like the one we have here, requires consistent steps to solve effectively.
Consider these general steps:

• Highlight what information is given and what needs solving.
• Determine relevant formulas and concepts (for example, using vector displacement and dot product for work calculations).
• Carry out calculations systematically.
• Double-check results to ensure accuracy.

Applying this structured approach allows us to break down complex problems. Considering units and symbols accurately throughout calculations avoids confusion. Whether in a classroom or in general science, these practices ensure you tackle physics questions with clarity and confidence.