How could you use a wrench to explain why a x b = - (b x a) for any vectors a and b?

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Think of a wrench tightening a nut on a bolt. Your arm puts a force clockwise on the end of the wrench and it rotates. This creates a torque on the nut, and since the nut is tightening, imagine that the torque vector points from the nut to the head...

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Think of a wrench tightening a nut on a bolt. Your arm puts a force clockwise on the end of the wrench and it rotates. This creates a torque on the nut, and since the nut is tightening, imagine that the torque vector points from the nut to the head of the bolt.

The equation for torque is t = r x F, where r is the length of the wrench and F is the force you're pulling on it with. To show that a = (bxc) = -(c x b), consider that the vectors F and r don't change direction or magnitude just because their order is switched in the equation. In other words, your arm is still tightening the nut, so the torque is still pointing in the same direction. So, you need a negative sign in front of  F x r to make t point in the same direction as at first.

t = r x F = -(F x r)

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