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Torque

  • Using the right-handed rule, we can see the orientation of three perpendicular vectors in a 3-dimensional space. However what these vectors actually mean has been more of a mystery to me, until I learned about torque.
  • Picture you have a wrench in your hand and you are looking to tighten a bolt.
    • The bolt is a vector τ\tau
    • The length of the wrench is a vector r\textbf{r} at a perpendicular angle
    • The angle the wrench will rotate at is in a direction F\textbf{F} that is perpendicular to both τ\tau and r\textbf{r}.
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  • In this case, F\textbf{F} is the force that is applied to r\textbf{r} on the wrench, which can be positioned as the cross product of these two vectors, equating to the torque τ\tau.
    • r×F=τ\textbf{r} \times \textbf{F}=\tau
  • Because the direction of the torque vector indicates the axis of rotation, we would represent the magnitude of τ\tau as follows:
    • ∣τ∣=∣r×F∣=∣r∣  ∣F∣  sin  θ|\tau|=|\textbf{r}\times\textbf{F}|=|\textbf{r}| \; |\textbf{F}| \; \text{sin} \; \theta
  • The magnitude of the torque is equal to the area of the parallelogram determined by r\textbf{r} and F\textbf{F}.