Two charges of # -6 C # and # 4 C# are positioned on a line at points # -6 # and # 9 #, respectively. What is the net force on a charge of # -1 C# at # 1 #?

You have to use the superposition principle. The net force is the vector sum of all the external forces acting on the -1C charge.

To calculate the electrical attractive/repulsive forces, we use Coulomb's Law. Coulomb's Law states that the electrostatic force is directly proportional to the charge of each object and has an inverse-squared relationship with the distance between the 2 charges.

Mathematically,

#F_"e" = frac{kQ_1Q_2}{r_12^2}#

#k ~~ 9xx10^9 "Nm"^2"C"^{-2}# is the constant of proportionality,
a.k.a. Coulomb's constant.

#Q_1# and #Q_2# are the charge of the 2 charges respectively.

#r_12# is the distance between the 2 charges.

From the formula, we can see that whether the force is attractive or repulsive depends on whether we are dealing with unlike charges or like charges.

In this question, I assume that the distance is measured in meters, since it is not specified.

The force exerted by the -6C at -6 is given by the coulomb law,

#F_1 = frac{k(-6"C")(-1"C")}{(7"m")^2}~~1.10xx10^9 "N"#

Since the unlike charges repel, the force is directed towards the right.

The force exerted by the 4C at 9 is given by the coulomb law,

#F_2= frac{k(4"C")(-1"C")}{(8"m")^2}~~-6.25xx10^7 "N"#

Since the like charges attract, the force is directed towards the right.

I recommend you to draw a diagram of the 3 charges to visualize it more clearly.

The sum of the 2 forces is #1.16xx10^9 "N"# to the right.