If resultant of two forces is given and smaller force is perpendicular to the resultant find both the forces.

Question

Two forces $\mathbf{F}_1$ and $\mathbf{F}_2$ such that $F_1$ is greater than $F_2$, the magnitude of their resultant force is $18$ N the resultant of this two forces is perpendicular to smaller one i.e $F_2$. find them?

Answer 1

You never can find any non-zero $\mathbf{F}_1, \mathbf{F}_2\in \mathbb{R}^{n}$ such that $\mathbf{F}_2 \perp (\mathbf{F}_1+\mathbf{F}_2)$

Considering,

\begin{align*} \mathbf{F}_1+\mathbf{F}_2 &= \mathbf{R} \\ (\mathbf{F}_1+\mathbf{F}_2)^2 &= R^2 \\ F_1^2+F_2^2+2\mathbf{F}_2 \cdot \mathbf{F}_1 &= R^2 \tag{1} \\ \mathbf{F}_2 \cdot (\mathbf{F}_1+\mathbf{F}_2) &= \mathbf{F}_2\cdot \mathbf{R} \\ \mathbf{F}_2 \cdot \mathbf{F}_1+F_2^2 &= 0 \tag{2} \end{align*}

Combining $(1)$ and $(2)$,

$$F_1^2 = R^2+F_2^2$$

From $(1)$, $$R^2 \ge F_1^2+F_2^2$$

contradiction!