Question

Could someone explain Torque and cross product:
T = r x F
picture is attached

Answer 1

Could someone explain the cross product in this picture. Specifically, how sin(theta) gives a perpendicular lever arm for the ladder and cos(theta) for the wall.

Answer 2

Or just a general explanation of the cross product would be helpful

Answer 3

The cross product is a vector operation between two vectors that produces a third vector that is perpendicular to the original two. This is means that a cross product is a three dimensional operation. The vector equation for torque is a cross product: \[\tau = r \times f\]where in this case the 'x' is the cross product symbol. According to the definition of a cross product this means that the torque vector is perpendicular to both the radius vector, r, and the force vector, f.

The cross product also follows the "right hand rule". To use the right rule (in this case in the torque example), extend your fingers on your right hand in the direction of the radius vector and then curl your fingers in the direction of the force vector. Now if you extend your thumb, it will point in the direction of the torque vector.

Another example would be to apply the right hand rule to the Cartesian coordinate axis: \[x \times y = z\]Extend the fingers of your right hand in the direction of the positive x-axis, then curl your fingers toward the positive y-axis. If you then extend your thumb it will point in the direction of the positive z-axis.

Also note that the magnitude of the cross product is given (again in terms of the torque equation) by the following:\[\tau=\left| r \right|*\left| f \right|\sin \theta\]Again, this is just the magnitude, not a vector. It's just the length of the vector without a direction.

You should do some reading on the right hand rule because it's used quite a lot in physics in mechanics, electromagnetic theory, and other areas as well.

Answer 4

Unfortunately, the best way to learn about the cross product is to read about it. Note that I also did not go into detail about the procedure for mathematically finding the cross product of two vectors. Instead, I refer you to the Wikipedia simple explanation: http://simple.wikipedia.org/wiki/Cross_product

Answer 5

Thank you. the sin vs cos was confusing me. I realize that to make the angle between the force of the wall and the radius it would have to be sin(90-theta) or cos(theta). Thank you.