2 forces F1 = 2i-3j and F2 = 3i + 12j are applied at a point. What force F must be applied at the point to counteract the resultant of these 2 forces?

R=F1+F2

F=-R=-F1-F2=-2i+3j-3i-12j combine terms.

To find the force F that counteracts the resultant of the two forces, we need to calculate the resultant force first.

The resultant force can be found by adding the two given forces F1 and F2. We simply add the corresponding components of the forces:

Fx = F1x + F2x = (2i - 3j)i + (3i + 12j)i = 2i^2 + 3i^2 + 12i^2 = 5i
Fy = F1y + F2y = (2i - 3j)(-3j) + (3i + 12j)j = -6ij - 9j^2 + 3ij + 12j^2 = -6ij + 3ij - 9j^2 + 12j^2 = -3ij + 3j^2

So the resultant force is F = 5i - 3ij + 3j^2.

To counteract this resultant force, a force with an equal magnitude but an opposite direction needs to be applied. Since the force F is also applied at the same point, the resultant force is canceled out and the net force becomes zero.

Therefore, the force F that needs to be applied at the point to counteract the resultant of the two forces is F = -5i + 3ij - 3j^2.