Calculate the Electric Field at the given point

[PyMaster]

Homework Statement

A -12 nC charge is located at the origin.

What is the electric field at the position (x2,y2)=(−5.0cm,−5.0cm)? Write electric field vector in component form.

Homework Equations

The Attempt at a Solution

Here, r^2 = (-0.05 - 0 )^2 meters squared. q=-12*10^-9 C.
For the unit vector, I chose \hat{i} and \hat{j} for the horizontal and vertical components of the vector. In the solutions manual, they choose a unit vector \frac {1} {\sqrt{2}}. I am not sure how they chose this as their unit vector.

I think they start with unit vectors of 5*\hat{i} and 5*\hat{j} then to convert to unit vectors they divide by the magnitude of each of the vectors (which is the length of the hypotenuse; 5*\sqrt(2).

Can someone verify this, please?

 

[Orodruin]

The point is that $\hat r$ should be a unit vector with the same direction as $\vec r$. The vector $\vec a = \hat i + \hat j$, this is not a unit vector since $\vec a^2 = 2$. Additionally, $\hat r$ should be pointing away from the origin, which in (-5cm, -5cm) means that it is given by $- \frac 1{\sqrt 2}(\hat i + \hat j)$.