Electric field problem, I got it right! But I don't know if the book and I have the s

Homework Statement

Three point charges are located on a circular arc as shown above. Find the electric force that would be on a -5.00nC point charge placed at P. Take r = 4.00cm

The Attempt at a Solution

The way I did it is to treat the top charge and bottom as the same, except for the vectors. I call the blue charge q2 and the 3.00nC as q1 and the -5.00nC as q3

$$k\frac{q_{1} q_{3}}{r^2}<cos30,sin30> + k\frac{q_{1}q_{3}}{r^2}<cos30, -sin30>$$ + $$k\frac{q_{2}q_{3}}{r^2}<1,0>$$

Simplifying

$$\frac{9q_{3}}{16 \times 10^-^4}<3\sqrt{3} - 2, 0>$$ = -8.98 x 105N/C

Which is exactly what the book had and I was very very happy, but I also held back

Concern I deducted that the -5.00nC is going to change the $$\vec_E$$ somehow, so I took the force as $$k\frac{q_{1}q_{3}}{r^2}$$ and then I secretly divided by 1nC as a "test charge". Now my question is, is my approach what the book wanted? Or did I get it by luck? Feel free to crush my spirit.

Thanks!

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