91Ó°ÊÓ

The dot product, also known as the scalar product, is a method used to multiply two vectors, resulting in a scalar (a single number). This mathematical operation plays a crucial role in physics, especially when calculating work using vectors.
The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is defined as:

• \( \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \)

This formula combines the corresponding components (i.e., x, y, and z) of the vectors. The result reflects the extent to which one vector projects onto the other.
If the vectors are perpendicular, the dot product is zero since they have no direct contribution to each other's direction.
In our exercise, the dot product of the force \( \vec{F} = (-5.00 \hat{\mathrm{i}} + 5.00 \hat{\mathrm{j}} + 4.00 \hat{\mathrm{k}}) \) and displacement \( \vec{d} = (2.00 \hat{\mathrm{i}} + 2.00 \hat{\mathrm{j}} + 7.00 \hat{\mathrm{k}}) \) gave us \(28.00\) J, indicating the total work done on the particle.

Kinetic energy is the energy that an object possesses due to its motion. It is an essential concept in mechanics and is expressed by the formula:

\( K = \frac{1}{2} mv^2 \)

Where \( m \) represents the mass and \( v \) is the velocity of the moving object.
In this formula, the velocity is squared, showing that kinetic energy is highly sensitive to changes in speed. If the velocity doubles, the kinetic energy increases by a factor of four.
In the context of the work-energy theorem, the work done on an object results in a change in its kinetic energy.
For our specific problem, we calculate both the initial and final kinetic energy to determine the particle's final speed, giving us an understanding of how the applied force affects its motion.
By starting with an initial kinetic energy of \( 16.00 \) J and adding \( 28.00 \) J of work, we achieve a final kinetic energy that allows us to calculate the new speed as \( 6.63 \mathrm{~m/s} \).

Displacement is a vector quantity that reflects the change in position of an object. Unlike distance, which is scalar and only accounts for the path length, displacement considers the direction and overall change in position.
Displacement is represented as:

• \( \vec{d} = x \hat{\mathrm{i}} + y \hat{\mathrm{j}} + z \hat{\mathrm{k}} \)

Where each component \( x \), \( y \), and \( z \) signifies the movement along the respective axes from an initial point to a final point.
Displacement can be positive or negative depending on the direction of motion. It provides a full picture of how far and in which direction an object has moved.
In this exercise, we are given a displacement vector \( (2.00 \hat{\mathrm{i}} + 2.00 \hat{\mathrm{j}} + 7.00 \hat{\mathrm{k}}) \mathrm{m} \), which tells us not only how far the particle has traveled from its initial point but also in what direction.
Understanding displacement is crucial for calculating the work done, as it determines how the force applied affects the motion of the particle.