# How the torque along an axis is defined, for any arbitrary body?

I recall torque about a point is defined as $$\vec T= \vec r \times \vec f$$

But my doubt is how do we define the torque about a general arbitrary axis, whose unit vector may be $$\hat{a}$$

My motivation for this question is that in all the cases taught to me the torque conveniently comes to be parallel to the axis vector, in other words the plane containing $$\vec F$$ and $$\vec R$$ comes to be perpendicular to the axis but what do I do when I encounter $$\vec F$$ and $$\vec R$$ to be arbitrary, in any direction. Finding rotational variables in that terms seems tricky.

I have no idea how to think here, and am not able to find any resources that explain this clearly with vectors and vector multiplication.

I did read somewhere that the component of the torque onto the axis itself is the torque along the axis, but there was no proof for this idea, and I don't have a clue how to write this vectorially (found in the coaching material for JEE exam, by ALLEN coaching institute)

Can someone prove this for me, or if I'm wrong about the previous statement, give me the correct formulation of torque with respect to an arbitrary axis?

Must it be that the radius vector at all times be perpendicular to the axis for the definition of torque to work? (Atleast in all the cases i have seen it has been so the case)

(Unrelated) my line of thinking :

I'm inclined to think in terms of vector triple product of the radius, force and the unit vector of the axis, and then representing it in the direction of the axis to preserve the vector form of the torque:

Torque about the axis $$\vec T_a = ((\vec R \times \vec F)\cdot\hat{a})\hat{a}$$

But I'm not sure if this is the right approach.

• As far as I know, torque is always defined about a point (by definition). Commented Nov 26, 2022 at 15:20
• Dear @user123 I have edited my question to include a PNG of the snippet of my book which discusses about the torque about an axis in brief. Check it out. (It is not a comprehensive book to learn from, it's just a huge problem book designed for JEE entrance exams, with a bit of theory in it) Commented Nov 26, 2022 at 16:04
• your torque equation is correct
– Eli
Commented Nov 26, 2022 at 16:05
• The last one I wrote? Can you link me some references if you have them so that I can cross verify my ideas? Commented Nov 26, 2022 at 16:12

the projection of the vector $$~\vec a~$$ towards $$~\vec b~$$ is
$$\vec a_p=a\,\cos(\phi)\,\hat b$$
with $$\vec a\cdot\vec b=a\,b\,\cos(\phi)\quad \Rightarrow\\ \cos(\phi)=\frac{\vec a\cdot\vec b}{a\,b}$$
$$\vec a_p=a\,\cos(\phi)\, \hat b=a\,\frac{\vec a\cdot\vec b}{a\,b}\,\hat b =(\vec a\cdot\hat b)\,\hat b$$