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Coulomb's Law fundamentally describes how charges interact with each other. It is the foundational principle explaining the force between two point charges. The formula, \( F = k \times \frac{{|q_1 \times q_2|}}{{r^2}} \), is straightforward yet potent.
Coulomb's constant \( k \) is approximately \( 8.99 \times 10^9\, \text{N} \cdot \text{m}^2/\text{C}^2 \), representing the force exerted per unit charge squared per square meter. It allows us to scale the magnitude of electrical forces to practical units.
Here, \( q_1 \) and \( q_2 \) are the charges involved, and \( r \) is the distance between them. This calculation tells us the force magnitude, which is always along the line joining the two charges, implying it is also inherently directional.

• Coulomb's Law helps us determine individual forces between charges to analyze their resultant effects when multiple charges act simultaneously.

In physics, understanding forces involves not just their magnitude but also their direction. This is where vector components come in. Every force acting in 2D or 3D space has parts that extend along each axis – most commonly, along the x and y-directions.
The concept of vector components allows decomposition of forces into more manageable parts. For instance, the force \( F_{13} \) exerted from the origin's charge to the point \( (3\, \text{cm}, 4\, \text{cm}) \) can be split into x- and y-components.
These components can be calculated as:

• The x-component, \( F_{13x} = F_{13} \cdot \frac{{3\, \text{cm}}}{{5\, \text{cm}}} \)
• The y-component, \( F_{13y} = F_{13} \cdot \frac{{4\, \text{cm}}}{{5\, \text{cm}}} \)

The fact that each vector component represents an aspect of the force in cases like this makes resolving the full force into its contributing directions essential to solve problems down to the axes of the coordinate system.

To fully characterize a two-dimensional force, one must identify both its magnitude and its direction. After determining the force components, combining them provides us with the complete picture.
The total force magnitude can be derived using the Pythagorean theorem. For our scenario, it is calculated as:

• \( F_{\text{TOTAL}} = \sqrt{ F_{\text{xTOTAL}}^2 + F_{\text{yTOTAL}}^2} \)

This value represents the overall effect of all contributing forces exerted on the charge.
Direction is equally important. It is generally expressed in degrees from a significant axis, typically the x-axis. Calculating this angle uses the tangent inverse function:

• \( \Theta = \tan^{-1} \left( \frac{F_{\text{yTOTAL}}}{F_{\text{xTOTAL}}} \right) \)

This direction aids in visualizing the force's inclination with respect to the positive x-direction, making interpretation in a coordinate system consistently intuitive.