• Scalar: A physical quantity completely described by its magnitude (a positive or negative number).
  • Examples: Mass, Volume, Time, Length, Density, Speed.
• Vector: A physical quantity that requires both magnitude and direction for complete description.
  • Examples: Force, Velocity, Acceleration, Position, Moment.

• Coplanar Forces: All forces in the system lie in the same 2D plane.
• Spatial (Non-Coplanar) Forces: Forces are distributed in 3D space and do not all lie in a single plane.
• Concurrent Forces: The lines of action of all forces intersect at a single, common point.
• Parallel Forces: All forces have parallel lines of action. They may be coplanar or spatial.
• Collinear Forces: A special case where all forces share the exact same line of action.
• General (Non-Concurrent, Non-Parallel) Forces: The most complex system, where lines of action neither intersect at a single point nor are parallel.

Identifying the type of force system is the first step in determining which equilibrium equations or resultant formulas are applicable.

• Equivalent Systems: Any force system acting on a rigid body can be replaced by an equivalent system consisting of a single resultant force $\mathbf{R}$ acting at a specific point $O$ and a resultant couple moment $\mathbf{M}_{RO}$. This simplifies the analysis of complex loading conditions on structural members like beams.
• Equivalent Wrench: In 3D space, it is often possible to further reduce an equivalent resultant force $\mathbf{R}$ and couple moment $\mathbf{M}_{RO}$ into a single force and a collinear couple moment. This specific combination is called a wrench. Its axis is called the wrench axis.

In three-dimensional space, couple moments are treated as free vectors that can be added vectorially.

Multiple couple moments can be added to form a resultant couple moment: $\mathbf{M}_R = \Sigma \mathbf{M}$. This is essential for reducing complex 3D loads into an equivalent single force and couple moment.

The dot product of two vectors yields a scalar value. It is particularly useful in Statics for finding the angle between two vectors or finding the projection of a vector onto a specific line.

Applications in Statics:

• Finding the angle between two vectors: $\theta = \cos^{-1}\left(\frac{\mathbf{A} \cdot \mathbf{B}}{AB}\right)$
• Finding the projection of a vector: The component of $\mathbf{A}$ parallel to a line defined by a unit vector $\mathbf{u}$ is $A_{\parallel} = \mathbf{A} \cdot \mathbf{u}$.

The direction of a 3D force vector $\mathbf{F}$ is defined by the coordinate direction angles $\alpha, \beta, \gamma$, measured between the tail of the vector and the positive $x, y, z$ axes, respectively. The cosine of these angles are known as direction cosines.

In 3D space, the moment of a force about a point $O$ is defined using the vector cross product. Unlike the dot product, the cross product yields a vector perpendicular to the plane containing the position vector and the force vector.

Sometimes we need to know the tendency of a force to cause rotation about a specific axis (like a door hinge) rather than a point. This is a scalar value, found using the mixed triple product.

• 2D Forces (Coplanar): All forces lie in a single plane (e.g., the $x$-$y$ plane). Forces are resolved into $x$ and $y$ components ($F_x = F \cos \theta$, $F_y = F \sin \theta$). The moment vector is always perpendicular to the plane (along the $z$-axis).
• 3D Forces (Spatial): Forces act in three-dimensional space. They are expressed using Cartesian vectors: $\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}$. The direction is defined by coordinate direction angles $\alpha, \beta, \gamma$, where $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.