For the resultant of several vectors, its x-component

Let the vectors be $\vec{A}_1, \vec{A}_2, \dots, \vec{A}_n$.

The resultant vector is $\vec{R} = \vec{A}_1 + \vec{A}_2 + \dots + \vec{A}_n$.

Taking the x-component on both sides, $R_x = A_{1x} + A_{2x} + \dots + A_{nx}$. Thus, the x-component of the resultant equals the sum of the x-components of the individual vectors.

The magnitude of the resultant satisfies $|\vec{R}| \le |\vec{A}_1| + |\vec{A}_2| + \dots + |\vec{A}_n|$.

Since the x-component of any vector cannot exceed its magnitude, $R_x \le |\vec{R}|$.

Combining these inequalities gives $R_x \le |\vec{A}_1| + |\vec{A}_2| + \dots + |\vec{A}_n|$.

If all vectors are directed along the positive x-axis, $R_x = |\vec{A}_1| + |\vec{A}_2| + \dots + |\vec{A}_n|$.

If the vectors are not all directed along the positive x-axis, $R_x < |\vec{A}_1| + |\vec{A}_2| + \dots + |\vec{A}_n|$.

Therefore, the x-component of the resultant might be equal to or less than the sum of the magnitudes of the vectors, but it can never be greater.