# Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that:

The moment of a force about a given axis (or Torque) is defined by the equation:

$M_X = (\vec r \times \vec F) \cdot \vec x \ \ \$ (or $\ \tau_x = (\vec r \times \vec F) \cdot \vec x \$)

But in my Physics class I saw:

$\vec M = \vec r \times \vec F \ \ \$ (or $\ \vec \tau = \vec r \times \vec F \$)

In the first formula, the torque is a triple product vector, that is, a scalar quantity. But in the second, it is a vector. So, torque (or moment of a force) is a scalar or a vector?

• The torque is a vector. In your $M_X$ you simply have the component of the torque along $\vec x$, and this component is a scalar. – ZeroTheHero Aug 2 '17 at 20:16
• Where have you seen the first version written? – sammy gerbil Aug 2 '17 at 21:29
• I think that its in Beer's Vectorial Mechanics for Engineers, but i'm not sure. – Vinicius ACP Aug 2 '17 at 21:50
• Technically, torque is a pseudo (or axial) vector, see en.wikipedia.org/wiki/…. – jim Jan 24 at 21:48

It is obviously a vector, as you can see in the 2nd formula.

What you are doing in the first one is getting the $x$-component of that vector. Rememebr that the scalar product is the projection of one vector over the other one's direction. Actually you should write $\hat{x}$ or $\vec{i}$ or $\hat{i}$ to denote that it is a unit vector. That's because a unit vector satisfies

$\vec{v}\cdot\hat{u}=|v| \cdot |1|\cdot \cos(\alpha)=v \cos(\alpha)$

and so it is the projection of the vector itself.

In conclusion, the moment is a vector, and the first formula is only catching one of its components, as noted by the subindex.

Torque (Force Moment) is a vector that describes the location of the Force line of action.

• Lemma: If you give me a force vector $${\vec F}$$ and a moment vector about the origin $${\vec M}$$ then I can define a line whose points obey the relationship $$\vec{M} = {\vec r} \times {\vec F}$$. This line has direction parallel to the force $${\vec F}$$ and passes through a point (closest to the origin) defined by $${\vec r} = \frac{ {\vec F} \times {\vec M} }{ \| {\vec F} \|^2 }$$

Proof: Use $$\vec{M} = {\vec r} \times {\vec F}$$ into the equation for the point.

$$\require{cancel} \frac{ {\vec F} \times {\vec M} }{ \| {\vec F} \|^2 } = \frac{ {\vec F} \times ({\vec r} \times {\vec F}) }{ \| {\vec F} \|^2 } = \frac{ \vec{r} ( \vec{F} \cdot \vec{F}) - \vec{F} (\cancel{\vec{F} \cdot \vec{r}} ) }{ \| {\vec F} \|^2 } = \vec{r} \frac{\| {\vec F} \|^2}{\| {\vec F} \|^2} = \vec{r}$$

This requires that $$\vec{F} \cdot \vec{r}=0$$ which is true for the point on the line closest to the origin.

It is true for both statics and dynamics that a moment is just a force at a distance. Only when the net force is zero (force couple) the moment is a pure moment and it does not convey any location information.

• @MichaelLevy - thanks. I changed the wording a bit. I hope you agree it is an improvement. – John Alexiou Jan 24 at 20:23
• @MichaelLevy - check the link. It is not working for me. – John Alexiou Jan 24 at 23:32
• @MichaelLevy Read also this post and my answer. – John Alexiou Jan 24 at 23:36
• The requested URL /metric/metric_public/vectors/vector_coordinate_geometry/vector_equation_of_line.html was not found on this server. – John Alexiou Jan 25 at 22:51
• where is your vector equation of a line? en.wikipedia.org/wiki/Line_(geometry)#As_a_vector_equation – Michael Levy Jan 25 at 23:00

There are some application, where we might want to quantify both the torque, which is a vector, and the component of the torque about a particular axis, which is a scalar.

I illustrate an example of this in the figure below, which is from 1 and provided here under fair use for the purpose of scholarship. The door is hinged so that it turns only around the $$\mathbf{\widehat{k}}$$ axis. Meanwhile, the door knob is located at a position $$\mathbf{r}$$ relative to the origin. A force $$\mathbf{F}$$ is applied to the door knob.

By $$\boldsymbol{\tau}$$, I denote the torque on the door knob, which is $$\boldsymbol{\tau} = \mathbf{r}\times \mathbf{F}.$$ By $$\tau_z$$, I denote the scalar component of the torque vector about the axis of rotation. So, $$\tau_z = \mathbf{\widehat{k}} \cdot \boldsymbol{\tau} = \left(\mathbf{r}\times \mathbf{F}\right)\cdot \mathbf{\widehat{k}}.$$

Bibliography

1 Mathematical Methods in the Physical Sciences, 3rd Edition, Mary L. Boas, ISBN: 978-0-471-19826-0 July 2005.