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Displacement plays a critical role in physics when analyzing the work done by a force. The displacement vector is a way of representing the change in position of an object.
In our example, the displacement vector is calculated by finding the difference between the initial and the final position of the object.

• The initial position is given as \( (1, 2, 0) \).
• The final position is \( (8, 4, 5) \).

To find the displacement vector, subtract the coordinates of the initial position from those of the final position. This results in a vector that points from the start to the end position.
In this case, the calculations are:

• Change in the x-direction: \( 8 - 1 = 7 \)
• Change in the y-direction: \( 4 - 2 = 2 \)
< li>Change in the z-direction: \( 5 - 0 = 5 \)

Thus, the displacement vector is \( \mathbf{d} = 7 \mathbf{i} + 2 \mathbf{j} + 5 \mathbf{k} \). This vector provides essential information required for calculating the work done.

The dot product is an algebraic operation used to find how much one vector aligns with another. It's a crucial element in calculating work because it combines both vectors: the force and the displacement.
The dot product between a force vector \( \mathbf{F} \) and a displacement vector \( \mathbf{d} \) determines the actual work done by breaking down the vectors into components and multiplying the respective components together.To find the dot product, follow these steps:

• Multiply the i-components of both vectors.
• Multiply the j-components of both vectors.
• Multiply the k-components of both vectors.

Then, sum up all these products. If given vectors are \ \mathbf{F} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k} \ \ and \ \mathbf{d} = 7\mathbf{i} + 2\mathbf{j} + 5\mathbf{k},\ The dot product is calculated as follows: - \( 2 \times 7 = 14 \) - \( 3 \times 2 = 6 \) - \(-1 \times 5 = -5 \)Add these products: \( 14 + 6 - 5 = 15 \). The result represents the work done in terms of the alignment of both vectors.

The force vector is a representation of the force applied to an object in three-dimensional space. Each vector component (usually represented in i, j, and k-terms) specifies the force's effective intensity along that particular axis.
In the context of calculating work done, the force vector determines how much force contributes to the movement of an object. In our example, the force vector is provided as \( \mathbf{F} = 2 \mathbf{i} + 3 \mathbf{j} - \mathbf{k} \). This means:

• Along the x-axis, a force of 2 units is applied.
• Along the y-axis, the force is 3 units.
• On the z-axis, there is a force of -1 unit, indicating it is acting in the opposite direction to motion.

The components allow us to understand how the force extends across spatial dimensions and affects the work done during the displacement of the object. Understanding how each vector component contributes is key to mastering the concept of work done.