Note: This section uses a coordinate plane (that's what you graph equations on, like graph paper) and the Pythagorean Theorem (used with right triangles). If you really want to know, basic trigonometry is used at the end, but it's not as important or even necessary for understanding the section. If you're a little shaky, read this section anyway. The next section is Free Body Diagrams and Equilibrium.
Just as a refresher:
• General Form for a Vector: Ai+Bj
To find what A and B are, there are two different methods, and both of them require a lot of math. You should be familiar with the coordinate-plane and right triangles for the first one and trigonometry for the second (don't worry if you don't know trigonometry, though. The second method is only a bonus and not crucial). For the first method, you need to know the coordinates of any two points along the line of the force, like in the example at right. You can then draw in two sides, parallel to the x and y axes, to form a triangle. These two sides represent the two component forces (introduced in The Parallelogram Law). Though their positions aren't exactly right (they should all be coming out of the same point of application), they're still the same length that they should be (ever looked at a parallelogram? Opposite sides are the same length). Notice that this is a right triangle? Good, because now we can use a slight variation of the Pythagorean Theorem, sometimes called the Distance Formula, to find the length between the two points.

So, we subtract the x-coordinates: 8-2=6, squared is 36. For the y-coordinates: 4-1=3, squared is 9. Added together, we get 45. The square root of this is approximately 6.708, the length between the two points. We then find the length of each leg of the triangle. The vertical leg has length 4-1=3 (its top is at (8,4) and its bottom is at (8,1)) since it's straight down from point A and straight across from point B). Using similar reasoning, the horizontal leg's length is 8-2=6. To find the relative length of each leg as compared to the hypotenuse (like the relative length of a component force as compared to the original), we divide each leg's length by the length of the hypotenuse. So, the vertical leg is 3/6.708, which is aproximately .447, and the horizontal leg is 6/6.708, or about .894.

So far, this is what we have: .894i+.447j. This is the unit vector.

Now we need to incorporate the magnitudes of the forces into this. So far, we haven't specified one, so let's say the original force was 100 Newtons. We know that the horizontal force is .894 as compared to the original, so the component's magnitude is .894*100, or 894 Newtons. The vertical force is .447*100, or 447 Newtons. Can you believe we're done? We can now rewrite our original force in terms of its x and y components, as a vector, in other words: 894i+447j. It's very easy to add two forces represented this way by a vector. Add the i parts together and then add the j parts together. As an example, we'll add a force, 500i+125j, to our other one. Their sum is (894+500)i+(447+125)j, or 1394i+572j. What could be simpler? Notice that now we're completely using math, and don't have to bother with the inaccuracies of trying to physically draw a parallelogram.

The second method (there's another one? Don't worry, it's not long) is a take-off of the first, though for it, you only need to know the angle that the original force forms with the x-axis instead of the coordinates of two points. To understand it, you'll also need to know about trigonometry (which isn't as bad as the name may sound).

If you know about trigonometry, you may have noticed that dividing the length of the vertical leg by the hypotenuse would be the same as taking the sine of Angle O (see example at right). Or that dividing the horizontal side by the hypotenuse would be the same as taking the cosine. After all, the definition of sine is "opposite over hypotenuse", and that of cosine is "adjacent over hypotenuse". You then proceed like in the first example. Angle O is 63.43 degrees, so the sine would be .894 and the cosine .447. Then, put them together, .894i+.447j, and you've got the unit vector.

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