Magnitude and Direction of Equilibrant

Theorem
Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space.

The equilibrant $\mathbf E$ of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ is:


 * $\mathbf E = -\displaystyle \sum_{k \mathop = 1}^n \mathbf F_k$

That is, the magnitude and direction of $\mathbf E$ is such as to balance out the effect of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$.

Proof
From Newton's First Law of Motion, the total force on $B$ must equal zero in order for $B$ to remain stationary.

That is, $\mathbf E$ must be such that:


 * $\mathbf E + \displaystyle \sum_{k \mathop = 1}^n \mathbf F_k = \bszero$

That is:


 * $\mathbf E = -\displaystyle \sum_{k \mathop = 1}^n \mathbf F_k$