Parallelogram of force

The parallelogram of forces is a method for solving (or visualizing) the results of applying two different forces to an object. It utilizes the principles of vector mathematics to solve the problem.

A vector is a mathematical device that has magnitude and direction. It is usually represented by an arrow. The direction that the arrow head faces is the direction, and the length of the arrow is proportional to the magnitude. The easiest way to visualize this is to think of a set of coordinate axes (your classic X-Y axes). Place an arrow shape with its tail resting on the origin (0,0) and its head pointing up the Y axis. The length of the arrow is proportional to the amount of force. If we were to place another arrow at the origin with its head facing to the right, along the X axis, this would represent a force operating at right angles to the first force. If this force were half as strong as the first force, the arrow would be half as long. For example we are going to push an object up the Y axis with a pressure (force) of 2 kilograms (or pounds). Someone else pushes the object to the right with a force of 1 kilogram. The object will obviously end up somewhere to the right of where we were pushing and somewhere above where the other people were pushing. In fact, the result will look like there was one force pushing midway between our two forces. You can find out what this apparent force looks like by putting the tail one arrow at the head of the other arrow. In our case (it really helps to draw this on a set of coordinate axes) one arrow extends from the origin (0,0) up the Y axis to the point 0,2. The smaller arrow went from 0,0 along the X axis to 1,0. We take the small arrow, and place its tail at the end of the larger arrow (0,2). The head of the smaller arrow now lies at 1,2. We would obtain the same result if we placed the tail of larger arrow on the head of the smaller arrow. This is called vector addition; you have summed the vectors. Now, if you draw a line from the origin (0,0) to the point where the sum of the vectors lies (1,2), you will represent the resultant - the force that results when you apply two different forces to an object. It's called a parallelogram because if you draw it both ways (i.e. put the small arrow at the tip of the large arrow, and put the large arrow at the tip of the small arrow you get a parallelogram, and the diagonal of the parallelogram is the resultant vector.

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