Resolving vectors

The components depend on your axes

A vector has a magnitude and direction. Its components describe it relative to axes you choose. Changing those axes changes the components, while the physical vector stays the same.

If a force of magnitude $F$ makes angle $\alpha$ with the positive $x$ axis, its components are

These expressions include signs through the angle. For simple right triangles, you can work out magnitudes first and then assign signs from the directions.

Let the triangle decide

“Horizontal means cosine” is not a general rule. The cosine belongs to the component adjacent to the angle you are actually using.

On a slope at angle $\theta$ above horizontal, weight has magnitude $mg\sin\theta$ parallel to the slope and $mg\cos\theta$ into the slope.

Two useful checks

• The squared components should add to the squared magnitude: $F_x^2 + F_y^2 = F^2$.
• At a simple limiting angle, does the component you expect to vanish actually become zero?

Components are replacements for a vector in your calculation, not additional physical forces.