91Ó°ÊÓ

In physics, understanding the dot product is crucial for computing work done by a force. Intuitively, the dot product gives a way of combining two vectors, which reflects how much one vector goes in the direction of another. This concept becomes particularly relevant when calculating work, as it encapsulates the interplay between force and motion.
The dot product of vectors \( \overrightarrow{\mathbf{F}} = (F_x \hat{\mathbf{i}} + F_y \hat{\mathbf{j}}) \) and \( \overrightarrow{\mathbf{d}} = (d_x \hat{\mathbf{i}} + d_y \hat{\mathbf{j}}) \) is given by:

• \( \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{d}} = F_x d_x + F_y d_y \)

This formula shows how the force components pair with their corresponding displacement components.
Alternatively, the dot product can be expressed in terms of the magnitudes of the vectors and the cosine of the angle \( \theta \) between them:

• \( W = F d \cos \theta \)

Here, \( F \) and \( d \) are the magnitudes of the force and displacement respectively. By accounting for the directionality of force, the dot product hence provides an effective tool to measure work.

Work requires both force and displacement. In this problem, we have a force vector and a displacement vector given in terms of their components. The force \( \overrightarrow{\mathbf{F}} = 2.0 \hat{\mathbf{i}} + 4.0 \hat{\mathbf{j}} \) N acts on an object that moves along a displacement \( \overrightarrow{\mathbf{d}} = 1.0 \hat{\mathbf{i}} + 5.0 \hat{\mathbf{j}} \) m.
Understanding these vectors means breaking them down in terms of their horizontal (i) and vertical (j) components. The work done by the force is calculated by evaluating the dot product of these vectors. It represents the energy transferred from the force to the object as it travels from one position to another. The closer the force is aligned with the direction of displacement, the more work it does. If the force were perpendicular to the displacement, the work done would be zero, as the dot product would be zero.
In simpler terms:

• Force and displacement must "point" in similar directions for work to be done.
• The magnitude of their alignment defines the work value.

Vectors in physics describe quantities with magnitude and direction. Force and displacement, both being vector quantities, can be described in terms of their components which lie along designated axes, usually represented as the i (horizontal) and j (vertical) axes.
For example, the force vector \( \overrightarrow{\mathbf{F}} = 2.0 \hat{\mathbf{i}} + 4.0 \hat{\mathbf{j}} \) N is split between its i and j components:

• i-component: 2.0 N
• j-component: 4.0 N

Similarly, the displacement vector \( \overrightarrow{\mathbf{d}} = 1.0 \hat{\mathbf{i}} + 5.0 \hat{\mathbf{j}} \) m is made of:

• i-component: 1.0 m
• j-component: 5.0 m

The key to using vector components effectively is understanding that the dot product formula essentially multiplies each component of force with the corresponding component of displacement. By doing this, we find how much of the force effectively 'does work' in the direction of each portion of the displacement, bridging the gap between the conceptual understanding and mathematical calculation of physics phenomena.