The moment of a force about a provided axis (or Torque) is characterized 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 mine 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, the is, a scalar quantity. However in the second, the is a vector. So, speak (or minute of a force) is a scalar or a vector?

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edited jan 25 in ~ 12:41

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asked Aug 2 "17 at 19:53

Vinicius ACPVinicius ACP
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It is clear a vector, as you have the right to see in the second formula.

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What you room 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. In reality you must write $\hatx$ or $\veci$ or $\hati$ to denote that the is a unit vector. That"s since a unit vector satisfies

$\vecv\cdot\hatu=|v| \cdot |1|\cdot \cos(\alpha)=v \cos(\alpha)$

and so the is the projection of the vector itself.

In conclusion, the minute is a vector, and the first formula is only capturing one that its components, as noted by the subindex.

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reply Aug 2 "17 in ~ 22:02

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

Lemma: If you give me a pressure vector $\vec F$ and a minute vector about the origin $\vec M$ then i can specify a line whose points obey the connection $\vecM = \vec r \times \vec F$. This line has actually direction parallel come the pressure $\vec F$ and also passes through a allude (closest to the origin) defined by $$\vec r = \frac \vec F \times \vec M \ $$

Proof:Use $\vecM = \vec r \times \vec F$ right into the equation for the point.

$$ \requirecancel \frac \vec F \times \vec M \vec F \ = \frac \vec F \times (\vec r \times \vec F) \ = \frac \vecr ( \vecF \cdot \vecF) - \vecF (\cancel\vecF \cdot \vecr ) \ = \vecr \frac^2 \vec F \ = \vecr $$

This calls for that $\vecF \cdot \vecr=0$ which is true for the suggest on the heat closest come the origin.

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It is true for both statics and dynamics that a minute is just a force in ~ a distance. Only as soon as the net force is zero (force couple) the moment is a pure moment and also it does no convey any kind of location information.