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In physics, the dot product is a mathematical operation that multiplies two vectors. This operation is particularly useful for calculating the work done by a force. The dot product of vectors \( \vec{A} = a_x\hat{i} + a_y\hat{j} \) and \( \vec{B} = b_x\hat{i} + b_y\hat{j} \) is calculated using the formula: \[ \vec{A} \cdot \vec{B} = a_xb_x + a_yb_y \] Here, we simply multiply the corresponding components of both vectors and then sum these products. This result helps us understand how the vectors relate in terms of their direction and magnitude.
When applied to force and displacement, the dot product effectively measures how much of the force is applied in the direction of the displacement, which is crucial for finding the work done by the force. Understanding and calculating the dot product is essential for solving many physics problems involving vectors.

Force \( \vec{F} \) and displacement \( \vec{d} \) are two fundamental vectors in physics. Force represents a push or pull on an object resulting from its interaction with another object. Displacement, on the other hand, represents a change in the position of an object. These two vectors can be represented as:
- Displacement: \[ \vec{d} = (8 \mathrm{~m})\hat{i} - (c \mathrm{~m})\hat{j} \]
- Force: \[ \vec{F} = (2 \mathrm{~N})\hat{i} - (4 \mathrm{~N})\hat{j} \]
Notice how each vector is broken down into its components along the i (horizontal) and j (vertical) directions. Calculating the work done by a force along a displacement involves analyzing how these directional components interact through the dot product. Also, it's important to recognize that changing the value of c in the displacement vector changes the resulting dot product, which affects the work calculation directly.

The concept of work done by a force in physics is defined as the energy transferred to or from an object via the application of force along a displacement. Mathematically, work \( W \) can be determined using the dot product of the force vector \( \vec{F} \) and the displacement vector \( \vec{d} \): \[ W = \vec{F} \cdot \vec{d} \] For our exercise, the calculation can be summarized as: \[ W = (2 \mathrm{~N}) (8 \mathrm{~m}) + (-4 \mathrm{~N}) (-c\mathrm{~m}) = 16 - 4c \] Breaking down the three cases:

• If the work done is zero, the forces are balanced, requiring \( c = 4 \).
• If work is positive, it suggests energy is being transferred to the particle, which happens when \( c < 4 \).
• If work is negative, energy is taken from the particle, occurring when \( c > 4 \).

Thus, recognizing how different values of c impact the work done helps understand the interaction between the forces acting on the particle and its resulting motion.