2.14:
Force Vector along a Line
Consider a three-dimensional static example like a pole with a cable anchored to the ground. Here, a tension force is acting along the cable.
Now, consider a Cartesian coordinate system with an arbitrary origin to represent the force as a Cartesian vector.
PA and PB represent the position vectors for the two ends of the cable. The force vector is directed from point A to B along the same direction as the position vector PAB. The unit vector specifies this direction.
Applying the triangle law of vector addition, the position vector along points A and B can be obtained by subtracting PA from PB.
Secondly, its magnitude can be found using the square root of the sum of the squares of its components.
Now, dividing the position vector by its magnitude provides the unit vector along the cable AB.
Finally, the product of the magnitude of the force vector and the unit vector expresses the force vector in the Cartesian form.
2.14:
Force Vector along a Line
Quite often in three-dimensional statics problems, the direction of a force is specified by two points through which its line of action passes. Consider a three-dimensional static pole with a cable anchored to the ground.
Considering a Cartesian coordinate system with the origin at the pole base, the endpoints of the cable can be denoted as A and B. PA and PB represent the position vectors for the two ends of the cable. The triangle law of vector addition is used to obtain the position vector along points A and B. For this purpose, the position vector PA is subtracted from PB.
The magnitude of the position vector can be obtained from the square root of the sum of the squares of its components.
The tension force acting on the cable is directed from point A toward point B following the same direction as the position vector PAB. The unit vector along the cable specifies the direction of the force. It is evaluated by dividing the position vector by its magnitude.
Lastly, the force vector can be expressed in the Cartesian form by multiplying the magnitude of the force vector and the unit vector. This product yields a three-dimensional vector representing the force acting on the pole.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 59