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A force vector describes the influence that produces a change in the motion of an object. It is a vector quantity, meaning it has both magnitude and direction. In physics problems, forces are often expressed in terms of their components along the coordinate axes, such as the x, y, and z axes.

For the given exercise, the force vector is defined as \( \vec{F}=(4.0 \,\text{N}) \hat{\text{i}}-(2.0 \,\text{N}) \hat{\text{j}}+(9.0 \,\text{N}) \hat{\text{k}} \).
This means the force has:

• A component of 4.0 N in the x-direction
• A component of -2.0 N in the y-direction
• A component of 9.0 N in the z-direction

Each component is associated with its axis-specific unit vector \( \hat{\text{i}} \), \( \hat{\text{j}} \), or \( \hat{\text{k}} \), indicating the direction of each component.
Understanding the components of a force vector is crucial as it allows us to consider the force's influence along each individual direction.

A velocity vector represents the speed and direction of an object’s motion. Like the force vector, it is also expressed in terms of its components along coordinate axes.

In the original problem, the velocity vector is given as \( \vec{v}=-(2.0 \,\text{m/s}) \hat{\text{i}}+(4.0 \,\text{m/s}) \hat{\text{k}} \).
This indicates:

• The object is moving at 2.0 m/s in the negative x-direction
• The object is moving at 4.0 m/s in the positive z-direction

Note that there is no y-component specified, suggesting no movement in the y-direction in the first scenario. Velocity vectors are essential in determining how different forces affect an object's motion, as they factor into the calculation of instantaneous power.

The dot product is a method to multiply two vectors, producing a scalar quantity. It is crucial in calculating work and power in physics because it inherently considers the direction of vectors.

For two vectors \( \vec{A} \) and \( \vec{B} \), where \( \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \), the dot product is calculated as:\[\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \]
In the problem at hand, the force and velocity vectors are used to find power:

• \( \vec{F} = (4.0 \text{N}) \hat{i} - (2.0 \text{N}) \hat{j} + (9.0 \text{N}) \hat{k} \)
• \( \vec{v} = -(2.0 \text{m/s}) \hat{i} + (4.0 \text{m/s}) \hat{k} \)

The dot product for the power calculation is:\[P = (4 \times -2) + (9 \times 4) = -8 + 36 = 28 \]Understanding the dot product helps us compute how aligned two vectors are, which is essential in understanding work done or power transferred when vectors are not perfectly aligned.

Power, in the context of physics, is the rate at which work is done or energy is transferred over time. When dealing with forces and motion, instantaneous power can be calculated using the dot product of the force and velocity vectors. This implies that power depends on how much of the force is acting in the direction of velocity.

In the first scenario of the problem, the power calculation is \[P = \vec{F} \cdot \vec{v} = 28 \, \text{W}\]This indicates the force is doing positive work on the object, leading to a power output of 28 Watts.

In the second scenario, we know the force is constant and power is given as -12 W with velocity having only a y-component, \( v_y \). Using the formula:\[P = -2 \cdot v_y = -12 \]Solving this gives:\[v_y = 6 \, \text{m/s} \] Power is an important concept as it gauges how effectively force displaces an object. Understanding power calculation allows one to infer energy dynamics in physical systems.