1. Three-Dimensional Forces Systems: Rectangular Components
Forces will now consist of all three components.
𝐹 = 𝐹 𝑥 𝑖 + 𝐹 𝑦 𝑗 + 𝐹 𝑧 𝑘
Directional cosines are used to determine the angle between the force and each of its components 𝑙=𝑐𝑜𝑠 𝜃 𝑥 , 𝑚=𝑐𝑜𝑠 𝜃 𝑦 , 𝑛=𝑐𝑜𝑠 𝜃 𝑧 𝐹 =𝐹𝑙 𝑖 +𝐹𝑚 𝑗 +𝐹𝑛 𝑘 =𝐹𝑐𝑜𝑠 𝜃 𝑥 𝑖 + F𝑐𝑜𝑠 𝜃 𝑦 𝑗 + 𝐹𝑐𝑜𝑠 𝜃 𝑧 𝑘
This is sometimes written in terms of its unit vector (unit vector in the direction of F 𝐹 =𝐹 𝑙 𝑖 +𝑚 𝑗 +𝑛 𝑘 =𝐹 𝑛 𝐹
Dot Product (Scalar Product)
This product of two vectors results in a scalar quantity. You multiply one vector by the component of the second vector that is parallel to the first vector.
If A = B: We use the same rules when multiplying a vector by itself.
The square of a vector only gives magnitude.

4. Three-Dimensional Forces Systems:
Moment and Couple
Forces and Moments are defined the same way in 3D as they were in 2D, except that there is a third component. It is typically easier to use vector notation to describe any 3D vector, especially for moments.
𝑀 is the net moment.
𝑀 𝑥 is the net moment in the x-dir (moment around x-axis).
𝑀 𝑦 is the net moment in the y-dir (moment around y-axis).
𝑀 𝑧 is the net moment in the z-dir (moment around z-axis).
𝑀 = 𝑀 𝑥 + 𝑀 𝑦 + 𝑀 𝑧
It is also useful at times to define the moment about a line (such as one of the coordinate axes) as opposed to a point.
Another techniques for determining the component of a moment is to dot the moment with the unit vector in the direction of one of the reference axis (does not have to be x, y or z). This results in a scalar quantity and must therefore be multiplied by the same unit vector to provide directional information.
𝑀 𝑛 = 𝑀 ∙ 𝑛 𝑛 = 𝑟 × 𝐹 ∙ 𝑛 𝑛
This is called the triple scalar product

5. 𝑀 𝑛 = 𝑟 × 𝐹 ∙ 𝑛 𝑛 = 𝑟 𝑥 𝑟 𝑦 𝑟 𝑧 𝐹 𝑥 𝐹 𝑦 𝐹 𝑧 𝛼 𝛽 𝛾 𝑛 𝛼= cos 𝜃 𝑥
𝑀 𝑛 = 𝑟 × 𝐹 ∙ 𝑛 𝑛 = 𝑟 𝑥 𝑟 𝑦 𝑟 𝑧 𝐹 𝑥 𝐹 𝑦 𝐹 𝑧 𝛼 𝛽 𝛾 𝑛
𝛼= cos 𝜃 𝑥
𝛽= cos 𝜃 𝑦
𝛾= cos 𝜃 𝑧
= 𝐹 𝑦 𝛾− 𝐹 𝑧 𝛽 𝑟 𝑥 − 𝐹 𝑥 𝛾− 𝐹 𝑧 𝛼 𝑟 𝑦 + 𝐹 𝑥 𝛽− 𝐹 𝑦 𝛼 𝑟 𝑧
= 𝑟 𝑥 𝐹 𝑦 𝛾− 𝑟 𝑥 𝐹 𝑧 𝛽 − 𝑟 𝑦 𝐹 𝑥 𝛾− 𝑟 𝑦 𝐹 𝑧 𝛼 + 𝑟 𝑧 𝐹 𝑥 𝛽− 𝑟 𝑧 𝐹 𝑦 𝛼
= 𝑀 𝑥𝑦 𝛾− 𝑀 𝑥𝑧 𝛽 − 𝑀 𝑦𝑥 𝛾− 𝑀 𝑦𝑧 𝛼 + 𝑀 𝑧𝑥 𝛽− 𝑀 𝑧𝑦 𝛼
= 𝑀 𝑦𝑧 − 𝑀 𝑧𝑦 𝛼− 𝑀 𝑥𝑧 − 𝑀 𝑧𝑥 𝛽+ 𝑀 𝑥𝑦 − 𝑀 𝑦𝑥 𝛾
= 𝑀 𝑦𝑧 − 𝑀 𝑧𝑦 cos 𝜃 𝑥 − 𝑀 𝑥𝑧 − 𝑀 𝑧𝑥 cos 𝜃 𝑦 + 𝑀 𝑥𝑦 − 𝑀 𝑦𝑥 cos 𝜃 𝑧
We have been using x, y and z here for convenience, but it could be applied to any reference frame!

6. Couples can still be used in 3D
Couples can still be used in 3D. Although when there are multiple couples the moment directions may not be parallel and therefore it is necessary to determine the net moment using vector calculus.
Similarly, it is possible to break a resultant moment into its components and determine the appropriate force couple necessary for each of the moment components.