The Component Method, Part 1 |
|||||||||

The component method of summing vectors is universally feared by introductory physics students, but is actually simple as long as you don't get too worried about trigonometric details. The foundation of the component method actually relies on a basic principle:
**The vectors point along the same line.**In this case, the sum of the two vectors is just the summation of the lengths if they point in the same direction, and the subtraction of the two lengths if they point in opposite directions.**The vectors are perpendicular to each other.**Here, simple mathematical relationships can be used to solve for the resultant vector.
To fully appreciate the importance of these two principles, complete the following web assignment. |
|||||||||

What are component vectors? Consider the equation vector We will take advantage of this property to greatly simplify vector
addition by replacing vectors that are hard to sum (we will call them Now there are any number of vectors that sum to vector But why not replace both bad vectors with good vectors that In the figure below, we have defined two directions with dashed
lines we call the x-direction (horizontal) and y-direction (vertical). If we
can somehow find replacement vectors for But how do we find these good vectors? |
|||||||||

Applet by Zona Land |
|||||||||

In the applet
above, we have isolated vector C (in black). Notice that vectors Now using the mouse, drag vector In the figure below, we have replaced both vectors We now have four vectors instead of two (see the figure below), but each of these four vectors is guaranteed to be either parallel or perpendicular to the others. Using our basic rules listed above, we can guess that we have now simplified the problem a great deal. And we have. |
|||||||||

A Little Bit of Trig (and I
Mean Little)Now that we know we have to replace But what are COS and SIN? Notice in the applet above that vectors But which do I choose? COS or SIN? The rule here is simple: Do you see the angle symbol that we have drawn in the figure? Well, if that angle sign touches the component vector for which you are trying to find its length, use COS.
If you use COS on one component vector, use SIN for the other. In summary, COS and SIN are "shrink factors" -- the amount of shrinkage depends on the angle. So in our example above, the length of vector B = C X COS(58), A = C X SIN(58).
Suppose that the length of vector B = 60 X COS(58) = 60 X 0.53 = 32, A = 17 X SIN(58) = 60 X 0.85 = 51.
According to these calculations, vector So all the trigonometry you are going to need at this point consists solely of finding
the angle symbol, seeing which component vector it touches, and using the COS button to
find the length of that vector. Use the SIN button for the other and Naturally, you would then do the same for vector Before moving on, test your ability to break a vector into its components. |
|||||||||