# Force Resultant is equal to the sum of the components -- why?

## Main Question or Discussion Point

Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?

berkeman
Mentor
Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
Why not? What else would they be equal to?

fresh_42
Mentor
Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
Because these are the addition laws of a vector space: $\begin{bmatrix}u_x\\u_y\end{bmatrix}+\begin{bmatrix}v_x\\v_y\end{bmatrix}=\begin{bmatrix}(u+v)_x\\(u+v)_y\end{bmatrix}\,.$ This coincides with the geometric vector addition of arrows. But whatever you take as a definition, they should yield the same result.

Tom.G

If one more person was added to the right end of the rope, which side would win?

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Pulling or pushing ?

Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
It follows from one of the axioms of Classical mech.: the Principle of Superposition