Write a reaction force as a vector in i and j form

Expert intro: Once you know that the wall’s force is equal in size and opposite in direction to R, the vector form is just the negative of R.

Detailed walkthrough

If the force exerted by the wall on the hook has the same magnitude as $\mathbf{R}$ but acts in the opposite direction, then the force vector is $\mathbf{F}=-\mathbf{R}.$

So if earlier you found $\mathbf{R}=ai+bj,$ then the wall’s force is $\mathbf{F}=-ai-bj.$

That is the cleanest way to write it using unit vectors $\mathbf{i}$ and $\mathbf{j}$.

💡 Pitfall guide

Don’t change only one component. The direction reverses completely, so every component must change sign. Also, if the original vector was written from the wall toward the hook, the reaction acts from the hook toward the wall.

🔄 Real-world variant

If the earlier answer was given in magnitude-angle form, first convert it into components before negating. For example, a force of magnitude $M$ at angle $\theta$ gives $M\cos\theta\,\mathbf{i}+M\sin\theta\,\mathbf{j}$, and the opposite force is the negative of that.

🔍 Related terms

unit vectors, reaction force, vector notation