Force Projections

A projection of a force onto an axis is an algebraic scalar quantity representing the component of the force vector along a selected direction. Projections convert vector relations into scalar equations and are therefore fundamental to analytical solutions in statics.

Projection onto a coordinate axis

The projection of a force $F$ onto the $x$ axis is determined from the angle $\alpha$ between the force vector and the positive direction of the axis:

$$F_x=F\cos\alpha$$

The sign of the projection is automatically accounted for by the cosine when the directed angle is used. If only an acute reference angle is given, determine the sign from the actual force direction.

If the force $\vec F$ forms an angle $\alpha$ with the positive $x$ direction, its projection is $F_x$. The projections $F_y$ and, in three-dimensional problems, $F_z$ are defined in the same way.

Signs of projections

A projection is positive when the corresponding force component points in the positive direction of the axis and negative when it points in the opposite direction. A force perpendicular to an axis has zero projection onto that axis.

Resolving a force in two dimensions

For a force in the $xy$ plane, $\vec F=F_x\vec i+F_y\vec j$. If $\alpha$ is measured from the positive $x$ axis, then $F_x=F\cos\alpha$ and $F_y=F\sin\alpha$, with signs determined by the actual component directions.

Checking the force magnitude

In a rectangular Cartesian coordinate system, the magnitude of a two-dimensional force satisfies $F=\sqrt{F_x^2+F_y^2}$. This relation is useful for checking calculated components.

Example

A force $F=10$ kN acts at $30^\circ$ above the positive $x$ axis in the first quadrant. Then $F_x=10\cos30^\circ\approx8.66$ kN and $F_y=10\sin30^\circ=5$ kN. Both projections are positive.

Common mistakes

• confusing an angle measured from the $x$ axis with one measured from the $y$ axis;
• using only the magnitude of a projection and ignoring its sign;
• using sine instead of cosine without checking which axis the angle is measured from;
• adding force magnitudes instead of algebraic force projections.

Equilibrium equations use the algebraic projections of all forces onto the selected coordinate axes.