How can a vector be resolved into components in various directions?

[sArGe99]

Homework Statement

I just want to get my concepts right again. How many times can a vector be resolved into horizontal and vertical components? In the case of a mass placed on a wedge, the weight of the body can be resolved into mg sin (angle) and mg cos(angle). Can this mg sin(angle) be further resolved and resolved again... Surely it would have components in every direction, then?

Homework Equations

Resolution using sine and cosine functions

The Attempt at a Solution

Again, none of the books really helped me on this

 

[LowlyPion]

Homework Statement

I just want to get my concepts right again. How many times can a vector be resolved into horizontal and vertical components? In the case of a mass placed on a wedge, the weight of the body can be resolved into mg sin (angle) and mg cos(angle). Can this mg sin(angle) be further resolved and resolved again... Surely it would have components in every direction, then?

Homework Equations

Resolution using sine and cosine functions

The Attempt at a Solution

Again, none of the books really helped me on this

Typically you want to choose a useful axis set. Gravity direction usually makes a fine choice and ⊥ planes make up useful 2nd and 3rd dimensions.

But generally you don't make a lot of different projections of things that are already projections since the original forces can be projected in whatever angles you need directly.

 

[sArGe99]

But in cases, like the one with an inclined plane the angle of inclination is given, it cannot be changed - say its theta. If we want the projection to be in other directions than what is given by sine and cosine components of theta, what do we do?
For example, mg sin theta acts parallel to the wedge surface and cosine component perpendicular. But what if we need a component parallel to the horizontal?

 

[LowlyPion]

But in cases, like the one with an inclined plane the angle of inclination is given, it cannot be changed - say its theta. If we want the projection to be in other directions than what is given by sine and cosine components of theta, what do we do?
For example, mg sin theta acts parallel to the wedge surface and cosine component perpendicular. But what if we need a component parallel to the horizontal?

I look to the original force wherever possible. Why take sines of cosines if you don't have to?