Cartesian Vector 3D to solve Force Vector 3D
Right Hand Coordinate System
A rectangular coordinate system is said to be right-handed if the thumb of the right hand points in the direction of the positive Z axis when the right-hand fingers are curled about this axis and directed from the positive x-axis towards the positive y-axis.
Rectangular Components of a Force vector 3D
A = AX + AY + AZ
A’ = AX + AY
A = A’ + AZ
When A is directed within an octant of the parallelogram law, we may resolve the vector into components A = A’ + AZ and then A’ = AX + AY
Cartesian Unit Vector for Force Vector (3D)
In 3D, the set of Cartesian unit vector i, j, k is used to designate the direction of the axis.
A = AX i + AY j + AZk
Coordinate Direction Angle
Cosα = Ax/A
Cosβ = Ay/A
Cosγ = Az/A
These are known as the direction cosines of A. Once they have been obtained, the coordinate direction angle α, β andγ can then be determined from the inverse cosines.
An easy way of obtaining these directions Cosines is to form a unit vector u in the direction of A. If A is expressed in cartesian vector form A = AX i + AY j + AZk then u will have a magnitude of one and it will be dimensionless.
u = Ax/A(i) + Ay/A(j) + Az/A(k)
Transverse and Azmuth Angles
The direction of A can be specified using two angles, namely a transverse angle ϑ and an azmuth angle.
you may visit the previous lecture for a better understanding of the topic.