Multiplying by unit vector in electric field equation

[ChrisBaker8]

Homework Statement

A 5.0 μC point charge is located at x = 1.0 m, y = 3.0 m, and a −4.0 μC point charge is located
at x = 2.0 m, y = −2.0 m.

a) Find the magnitude and direction of the electric field at x = −3.0 m, y = 1.0 m. [5 Marks]
b) Find the magnitude and direction of the force on a proton at this position in a).

Homework Equations

E = (kq/r²) x [r-hat] (couldn't find the symbol for r circumflex)

The Attempt at a Solution

I can work out the first part of the equation, but I'm unsure what r-hat represents and what figure to use. I thought a unit vector was in the form Ai + Bj + Ck, how do you multiply that by a figure to get another single figure?

 

[rl.bhat]

Let point A be ( 1 m, 3 m) and point B be (2 m, -2 m). Let P be the point (-3 m, 1 m) where you want to find the electric field.
First of all find the distances AP and BP. Finding slope of the line AP and BP, you an find the angle made by the electric fields with x-axis. Then resolve them along vertical and horizontal components. Then find ΣEx and ΣEy and E.

 

[kerimek]

As a scientist, it is important to understand the concept of unit vectors in relation to electric fields. In the electric field equation, the unit vector [r-hat] represents the direction of the electric field at a specific point. It is a vector with a magnitude of 1 and points in the direction of the electric field.

To answer the first part of the question, you can use the given equation and plug in the values for the charges and their positions. Then, you can use the properties of unit vectors to find the direction of the electric field at the given point. The magnitude of the electric field can be calculated using the distance between the point charges and the distance between the point charge and the given point.

For the second part of the question, you can use the equation F = qE, where q is the charge of a proton and E is the electric field calculated in part a). This will give you the magnitude of the force on the proton. To find the direction of the force, you can use the right-hand rule, where the direction of the force is perpendicular to both the electric field and the velocity of the proton.

It is important to note that the unit vector [r-hat] is used to represent the direction of the electric field, not to multiply by a figure to get another single figure. It is a mathematical tool used to simplify calculations and represent the direction of a vector. I hope this helps clarify the concept of unit vectors in the electric field equation.