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
Just as a refresher:
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.
- General Form for a Vector: Ai+Bj
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
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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Body Diagrams and Equilibrium