Given that 𝐹 one, 𝐹 two, and 𝐹 three are three coplanar forces in equilibrium meeting at a point where 𝐹 one equals five 𝑖 minus three 𝑗 and 𝐹 two equals four 𝑖 minus 14𝑗, find 𝐹 three.
As the three forces are in equilibrium, 𝐹 one plus 𝐹 two plus 𝐹 three must be equal to zero 𝑖 plus zero 𝑗. Substituting in the values of force, 𝐹 one and 𝐹 two, gives us five 𝑖 minus three 𝑗 plus four 𝑖 minus 14𝑗 plus 𝐹 three equals zero 𝑖 plus zero 𝑗.
Grouping the like terms on the left-hand side gives us nine 𝑖 minus 17𝑗 plus 𝐹 three is equal to zero 𝑖 plus zero 𝑗, as five 𝑖 plus four 𝑖 is equal to nine 𝑖 and negative three 𝑗 minus 14𝑗 is negative 17𝑗.
Balancing this equation by subtracting nine 𝑖 and adding 17𝑗 to both sides of the equation gives us 𝐹 three is equal to negative nine 𝑖 plus 17𝑗. This means that, in order for the three coplanar forces to be in equilibrium, where 𝐹 one equals five 𝑖 minus three 𝑗 and 𝐹 two equals four 𝑖 minus 14𝑗, then 𝐹 three must be equal to negative nine 𝑖 plus 17𝑗.