The moment of a force: turning effect, coordinates and Varignon

Drawing an arm is not always easy, so we use coordinates. Put the moment centre at the origin O. A force F = (F_x, F_y) acts at a point (x, y) of its line of action. Split it into its two parts. The vertical part F_y acts at horizontal distance x from O, so its arm is x and its moment is x F_y: an upward force to the right of O turns anticlockwise (positive), an upward force to the left turns clockwise (negative), and the sign of x does exactly that. The horizontal part F_x acts at height y above O, and a rightward force above O turns clockwise, so its moment is −y F_x. Adding the two gives M = x F_y − y F_x. This number is positive for anticlockwise turning and negative for clockwise. You can check that it does not matter which point of the line of action you take: moving the point by t along the force changes M by t F_x F_y − t F_y F_x = 0. If the moment centre is another point B, replace (x, y) by the position of the point of action measured from B, that is A − B. Once M is known, the arm is L = |M| / F, where F = √(F_x² + F_y²). (Pure Mathematics will soon write this same number as r × F; you do not need that yet.)