AMME1802 Chap.3 Three-Dimensional Force Systems
Three-Dimensional Force Systems
Spatial force problems become a sequence of coordinate operations: form a difference vector, normalise it, apply the force magnitude, and then choose a dot or cross product. This chapter covers direction cosines, scalar projections, moments and moments about an axis, with endpoint order and unit consistency treated as part of the method.
Points play different roles: cable endpoints establish force direction, while a moment centre and application point establish the lever arm. Mixing those pairs produces plausible determinants with the wrong geometry. Direction cosines preserve signs, so a negative component correctly produces an obtuse angle from the corresponding positive axis.
A scalar projection uses a dot product and can be negative; a moment uses a cross product and remains a vector until an axis projection is requested. Unit-vector length, reconstructed force magnitude and force-times-length dimensions form a compact audit set.
What this chapter covers
- 01
Coordinate difference vectors
- 02
Direction cosines
- 03
Scalar projections
- 04
Three-dimensional moments
- 05
Moment components about axes
- 06
Geometric tests for parallel and perpendicular vectors
- 07
Perpendicular-distance checks on cross-product moment magnitudes
- 08
A final spatial audit using units, magnitudes and right-hand direction
Worked example · free
Cable force and axis moment
- +1Subtract cable endpoint coordinates in the physical force direction.
- +1Normalise the difference vector and multiply by tension.
- +1Build the position vector from the moment centre to the application point.
- +1Calculate r cross F for the moment vector.
- +1Dot that moment with the axis unit vector and interpret its sign.
Key terms
- Direction cosine
- A unit-vector component equal to the cosine of the angle from a positive axis.
- Scalar projection
- The signed amount of a vector along a specified unit direction.
- Axis moment
- The scalar component of a moment vector along a stated axis.
- Position vector
- A directed segment from a chosen origin or moment centre to a point.
- Axis projection
- The signed component of a vector along a specified axis.
- Right-hand rule
- The convention connecting cross-product order to the positive perpendicular direction.
- Lever arm
- The perpendicular separation between a moment centre and a force line.
- Oblique axis
- An axis that does not coincide with a global Cartesian coordinate direction.
Three-Dimensional Force Systems FAQ
Why can a direction angle be obtuse?
A negative vector component means the vector points partly opposite the corresponding positive axis. Its angle from that positive axis therefore exceeds ninety degrees. Replacing it with an acute reference angle reverses the represented direction.
What is the difference between dot and cross products?
A dot product returns a scalar amount along a direction, such as a force projection. A cross product returns a vector perpendicular to both inputs, such as a moment. Match the output type to the quantity requested.
How do I check a three-dimensional force?
Square and sum the unit-vector components to confirm they total one, then reconstruct the force magnitude from its Cartesian components. For moments, confirm force times length units and verify the cross-product direction with the right-hand rule.
How is a moment about an oblique axis found?
Calculate the full moment vector about any point on the axis, construct a unit vector in the declared positive axis direction and take their dot product. The result is signed, and reversing the axis direction reverses that sign.
Assessment move
Write labelled endpoint vectors on separate lines before any determinant. Drill direction-cosine questions until obtuse angles feel normal, then alternate scalar projections with moments. Use magnitude reconstruction and unit-vector length as automatic checks on every page.
Create a fixed coordinate workflow in your notes: list point coordinates, write the named difference vector, compute its magnitude, form the unit vector and multiply by force. Only after that chain should you open a dot or cross product. Practise one problem where the projection is negative so the sign has a physical meaning, and one where a direction angle exceeds ninety degrees.
For moments, draw r from the selected centre to any point on the force line and predict which coordinate plane contains the moment. When an axis moment is requested, calculate the full moment first and then project it; do not select one Cartesian component unless the requested axis actually matches it. Add a unit conversion check before every determinant.
Finish each practice set by testing n dot n, rebuilding the force magnitude and comparing the implied perpendicular lever arm with the coordinate dimensions. Create coordinate cards that give four points and ask for several differently ordered vectors, such as AB, BA and CD. This makes endpoint order deliberate before forces are introduced.
Next assign a tension magnitude and calculate its direction angles, checking that the cosine squares sum to one. Alternate dot-product questions with cross-product questions and write the expected output units before calculating. Add an oblique-axis moment in which the requested axis is not a coordinate axis, forcing the full moment vector to be projected.
Sketch the right-hand direction for each cross product and compare it with the determinant sign. A final mixed problem should use one point pair for cable direction and another point pair for a moment axis so their roles cannot be confused.
