Name: calculating moment with cartesian vectors
Uploaded: 2025-05-22T09:06:02.000Z
Duration: 1 min 26 s
Channel: JoVE

The moment of force is defined as the cross-product of position and force vectors.
The cross-product for the moment can be expressed in the determinant form, using unit vectors and the Cartesian form of position and force vectors.
The determinant is expanded to determine the Cartesian vector formulation of the moment of force.
Consider that a force of 10 N is applied to a revolving door at a point given by the position vector in the cartesian components form. Here, the moment of force in the vector formulation needs to be determined.
In this case, the force vector is perpendicular to the y-z plane , which results in the y and z components of the force vector being zero. A determinant is constructed using the position vector and the force vector.
Finally, by expanding the determinant, the revolving door&#39;s moment of the force about the z-axis is determined.
Here, the moment is in the positive z-direction, implying the door rotates counterclockwise.

cartesian form for vector formulation

The Cartesian form for vector formulation is a process to calculate  the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.
<img alt="Mo matrix for static equilibrium, vector cross product, equation, physics diagram." src="/files/ftp_upload/14254/14254_%20Equation_1v2.svg" style="text-align: center; height: 60px;" data-altupdatedat="1754904545000" data-alt="static equilibrium, vector cross product Mo formula, determinant matrix diagram, physics education">
The determinant can then be expanded to determine the Cartesian vector formulation of the moment of the force.
<img alt="Static equilibrium equation; vector moment notation; ΣMO=0; educational diagram; mechanics concept." src="/files/ftp_upload/14254/14254_%20Equation_2v2.svg" style="text-align: center; height: 15px;" data-altupdatedat="1754904545000" data-alt="3D Moment equation diagram, showcasing static equilibrium principle: \( M_O = (r_y F_z - r_z F_y)i - (r_x F_z - r_z F_x)j + (r_x F_y - r_y F_x)k \).">
For example, suppose a force of 10 N is applied to a revolving door along the x-direction at a point given by the position vector r = 0i - 4j + 0k in the Cartesian component form. In this case, the force vector is perpendicular to the y-z plane along which the door stands, which means that the y and z components of the force vector are zero. To calculate the moment of force, a determinant is constructed using the position vector and the force vector, F = 10i + 0j + 0k.
<img alt="Matrix determinant, static equilibrium equation, ΣMO=0, featuring vector components (i, j, k)." src="/files/ftp_upload/14254/14254_%20Equation_3v2.svg" style="text-align: center; height: 60px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia matrix representation with vectors i, j, k; static equilibrium equation.">
Finally, by expanding the determinant, we can determine that the revolving door's moment of force equals 40 N.m about the z-axis. The positive moment implies that the door rotates counterclockwise.
The Cartesian form for vector formulation allows us to calculate the moment of force using position and force vectors in a straightforward manner.

Cartesian Form for Vector Formulation

The Cartesian form for vector formulation is a process to calculate&#160; the moment of force using the position and force vectors. The moment of force is ...

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calculating moment with cartesian vectors

Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form.

The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step.

A revolving door example shows the process in action. A force of 10 N is applied in the x-direction at the position vector r = 0 i - 4 j + 0 k. The force vector has no y or z components, so it can be written as F = 10 i + 0 j + 0 k.

Using these vectors, the determinant is built and expanded. The result is a moment of 40 N.m about the z-axis. The positive sign means the door rotates counterclockwise.

Calculating Moment with Cartesian Vectors

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Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form. The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step. A revolving door...

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key terms and concepts

Learn how Cartesian vector form is used to calculate moment of force with unit vectors, determinants, and a revolving door example in a simple step-by-step method.

Calculating Moment with Cartesian Vectors | JoVE K12

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Calculating Moment with Cartesian Vectors

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