Decomposition of Vectors: Analyzing the General Case

General Case

So it is this vector. Why? Well, if we consider the right triangle formed by this vector, we
can read off its components from this right triangle.
We begin by noting that the length of w is equal to the length of v. So the length of w is v2.
This side is v2, the length of w. What about this side? The cosine of theta is v2 on this
side, and the sine of theta is v2 on this other side.

The vertical component of velocity can be calculated as v2 cosine theta, where v is the
magnitude of the velocity vector and theta is the angle between v and the horizontal axis.
The horizontal component is found by calculating v2 sine of theta, which has a negative
sign because it is in the x direction and not towards or away from you.
Good, those were warmups. In general, v is just pointing in some arbitrary direction. So the
general case is more complicated than those two, we don't immediately see how to do it.

If you just look at this picture, it's hard to do. There is a trick to figuring out what happens
to a general v. We will draw a rectangle around v and imagine rotating the whole rectangle
instead of just rotating v.
This is an example of what a before picture looks like.
So, in this case we will write v. And we are going to think of v, let's draw a rectangular
frame around v.
The bottom of the frame is blue, red, and red. OK. Now imagine rotating this picture with
the "v" shapes sitting inside a rectangular frame of color.
After rotating the rectangle by theta radians, the figure will appear as shown here. Theta is
the angle by which we rotated the rectangle: w is our new vector of interest.
This image makes sense.
Now, in the before picture, v is v1, comma v2. The blue vector is v1 comma 0. The red
vector is 0 comma v2. Let's look at this for a second
The blue vector is v1 comma O, and it's in the x direction. The red vector is O comma v2,
and it's in the y direction. There are two red vectors here (they have the same length), but
you can see that they're both going straight up.
Thus, they are both O comma v2. We will now explore the after picture.
So the first question for you is: Are the two red vectors in the after picture the same vector
or not?
Thoughts can be divided by halves.
Think over. We began with a rectangular box, rotated it by angle theta to get this figure,
and found two vectors in the picture. Are they the same? Try to solve it
[Graph][Graph][Graph]