A charge \(q\) is at the point \(x=1 \mathrm{m}, y=0 \mathrm{m} .\) Write expressions for the unit vectors you would use in Coulomb's law if you were finding the force that \(q\) exerts on other charges located at (a) \(x=1 \mathrm{m}, y=1 \mathrm{m} ;\) (b) the origin; and \((\mathrm{c}) x=2 \mathrm{m}, y=3 \mathrm{m}\) You're not given the sign of \(q .\) Why doesn't this matter?

Electric charge is a fundamental property of matter that leads to electrostatic interactions. It exists in two types, commonly referred to as positive and negative charges. Objects with opposite charges attract each other, while those with similar charges repel.

Encoded in the very nature of electric charge is the principle that like charges repel and unlike charges attract. This is a critical concept to grasp when studying Coulomb's Law and its applications. Charge is measured in Coulombs (C), and the amount of charge affects the magnitude of the electrostatic force experienced by objects.

Importantly, the sign of the charge (+ or -) determines the direction of the force (attractive or repulsive), but in the context of finding unit vectors, as in the provided exercise, the sign is not crucial because unit vectors only indicate direction, not magnitude or type of force.

A unit vector is essentially a direction pointer. It has a magnitude of one and points in a specific direction. When dealing with forces such as the electrostatic force, the unit vector plays an essential role in determining the direction of the force.

In electrostatic problems, we often calculate the force exerted by one charge on another. The unit vector helps us express this force in vector form—showing not just how strong the force is, but also where it's going. It's obtained by dividing a displacement vector, which points from one charge to the location of another, by its magnitude.

This concept is critical for visualizing electrostatic forces in space and is particularly useful when working with Coulomb's Law. By understanding unit vectors, students can better grasp the vector nature of forces and solve for them in multidimensional spaces.

The electrostatic force, sometimes simply called an electric force, is a force that occurs due to the presence of electric charge. Coulomb's Law provides a mathematical equation which quantifies the electrostatic force between two charges. The law states that the magnitude of the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Coulomb's Law is fundamental to electrostatics and can be expressed as: \[\begin{equation}F = k \frac{|q_1q_2|}{r^2}\end{equation}\], where \(F\) is the force magnitude, \(k\) is Coulomb's constant, \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between the charges. In the context of the provided exercise, understanding how to compute the direction of the electrostatic force using the unit vector is crucial, as this provides a deeper understanding of how charged particles interact in space.