There are many situations that require that. See below for examples.
- An object on an incline with angle
#theta#with respect to horizontal:
The weight of the object tends to accelerate it down the incline but the friction of the object with the incline tends to hold it still. Which wins? If the object does move, it will be along the incline. So we need to compare the forces whose directions are up or down the incline.
The weight is a vertical force. Only the component of the weight along the incline (
#m*g*sintheta#) would tend to make it move down the incline.
Calculating friction requires the normal force, N, which presses the 2 surfaces together. This force is the component of the weight that is perpendicular to the incline (
- A swimmer crossing a river:
The swimmer's effort is directed straight across the river. But the river's current takes her downriver. If you can calculate the actual length of her path across the river and the time it took, you can calculate her speed along that path diagonally across the river. If you also know her speed in still water, can you calculate the speed of the river's current?
- And there are many more situations that call for resolving a vector.
I hope this helps,