Name: finding a cable force from two points
Uploaded: 2025-05-22T09:06:01.000Z
Duration: 1 min 26 s
Channel: JoVE

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.

force vector along a line

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.
<img alt="3D force vector diagram, static equilibrium concept, labeled points A(xA,yA,zA) and B(xB,yB,zB)." src="/files/ftp_upload/14237/14237_PT_Fig_1v2.svg" style="text-align: center; height: 350px;" data-altupdatedat="1754904545000" data-alt="3D force vector diagram with coordinates; shows equilibrium on axes with points A and B.">
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.
<img alt="Vector equation diagram; position vector formula \(P_{AB}\) with components in 3D space for calculations." src="/files/ftp_upload/14237/14237_PT_Equation_1v2.svg" style="text-align: center; height: 15px;" data-altupdatedat="1754904545000" data-alt="Vector calculation formula \( \textbf{P}_{AB} \) in 3D space with unit vectors i, j, k.">
The magnitude of the position vector can be obtained from the square root of the sum of the squares of its components.
<img alt="Distance formula equation \( P_{AB} \) for spatial points; mathematical concept analysis diagram." src="/files/ftp_upload/14237/14237_PT_Equation_2v2.svg" style="text-align: center; height: 26px;" data-altupdatedat="1754904545000" data-alt="Distance formula equation P&lt;sub&gt;AB&lt;/sub&gt;=√((x&lt;sub&gt;B&lt;/sub&gt;-x&lt;sub&gt;A&lt;/sub&gt;)²+(y&lt;sub&gt;B&lt;/sub&gt;-y&lt;sub&gt;A&lt;/sub&gt;)²+(z&lt;sub&gt;B&lt;/sub&gt;-z&lt;sub&gt;A&lt;/sub&gt;)²).">
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.
<img alt="Unit vector equation in 3D coordinate system, representing direction magnitude calculation." src="/files/ftp_upload/14237/14237_PT_Equation_3v2.svg" style="text-align: center; height: 39px;" data-altupdatedat="1754904545000" data-alt="Vector direction formula, unit vector equation, 3D coordinates diagram, physics or engineering concept.">
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.
<img alt="Force equilibrium equation F=Fu; mathematical formula diagram for static equilibrium analysis." src="/files/ftp_upload/14237/14237_PT_Equation_4v2.svg" style="text-align: center; height: 11px;" data-altupdatedat="1754904545000" data-alt="Force equation, F=Fu, formula illustration, educational physics concept, static equilibrium.">

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 ...

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finding a cable force from two points

A cable force can be found from the two points that define its line of action. In three-dimensional statics, this setup often appears with a pole and a cable anchored to the ground. The cable ends are labeled A and B, and their position vectors are written as P_A and P_B.

The position vector from A to B is found using the triangle law of vector addition. First, subtract P_A from P_B to get P_AB. Then find the magnitude of P_AB by taking the square root of the sum of the squares of its components. This gives the length of the vector in Cartesian form.

The tension force in the cable points from A toward B. It has the same direction as the position vector P_AB. To show that direction, divide P_AB by its magnitude to get the unit vector along the cable. A unit vector is a vector with length 1 that gives direction only.

The force vector is written in Cartesian form by multiplying the force magnitude by the unit vector. This product gives a three-dimensional vector for the force acting on the pole. The method connects the geometry of points A and B to the final force vector.

Finding a Cable Force from Two Points

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Learn how to use two position vectors, a unit vector, and force magnitude to express a cable’s tension as a 3D Cartesian force vector.

Finding a Cable Force from Two Points | JoVE K12

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Finding a Cable Force from Two Points

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