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Resolving Forces

Decomposing a force into components parallel and perpendicular to a given plane, facilitating force sum calculations.
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The statement of the theorem

Let F⃗∈R3\vec{F} \in \mathbb{R}^3 be a force vector acting on a point in a Cartesian coordinate system defined by the orthonormal basis B=(i^,j^,k^)B = (\hat{i}, \hat{j}, \hat{k}). The process of resolving F⃗\vec{F} involves decomposing it into its orthogonal components relative to this basis. This decomposition is formally expressed as the unique linear combination:\n\nF⃗=Fxi^+Fyj^+Fzk^\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}\n\nwhere Fx,Fy,Fz∈RF_x, F_y, F_z \in \mathbb{R} are the scalar components of the force along the x,y,zx, y, z axes, respectively. These components are determined by the projection of F⃗\vec{F} onto each basis vector, utilizing the dot product operator (⋅)(\cdot): \n\nFx=F⃗⋅i^=∑k=13Fkδk,xF_x = \vec{F} \cdot \hat{i} = \sum_{k=1}^{3} F_k \delta_{k, x} \n\nIn general, for an arbitrary orthonormal basis B′=(u^1,u^2,u^3)B' = (\hat{u}_1, \hat{u}_2, \hat{u}_3), the components (F1′,F2′,F3′)(F'_1, F'_2, F'_3) are given by:\n\nFi′=F⃗⋅u^i, for i∈{1,2,3}F'_i = \vec{F} \cdot \hat{u}_i, \text{ for } i \in \{1, 2, 3\}
Source: Wikipedia