Why Howard, this is easy, if only you use quaternions*

Define the windward direction to be y and down to be z (use your favorite version of the right hand rule to find x :

http://xkcd.com/199/). So the half-angle rotation vector of your personal pointing vector to the vector referenced by this projection is

r_g = cos(22.5) + k * sin(22.5)

These will be raised up by 45 degrees from the vertical, so in a reference frame that uses r_va as it's x direction, the half-angle rotation vector from you to the plane will be

r_u = cos(22.5) + j * cos(22.5)

To find the vector from one to one's plane, one need merely multiply r_g * r_u * i * r_u' * r_g' (using the well-known** rules of quaternion multiplication), to get

i / 2 - j / 2 - k / sqrt(2)

That's a nice check, and gives us the expected results (note that the first time I did this I inexplicable forgot the order of multiplication for the rotations -- and of course, since quaternions are not commutative, the answer to r_u * r_g * i * r_g' * r_u' is different).

Of course, the plane's direction of flight is in a frame of reference that's rotated 90 degrees from the lines:

r_p = cos(45) - k * sin(45)

(as a check, in level flight the plane's 'nose' vector is r_p * i * r_p' = -j -- that's exactly as expected).

So the airplane's pointing vector with respect to the earth is simply

r_g * r_u * r_p * i * r_p' * r_u' * r_g' = (-i - j)/sqrt(2)

Dang. Well, that's a really good check, but you wanted the

*wind's* pointing vector with respect to the

*plane*, didn't you. Oh well -- just do the rotations in the reverse direction and the reverse order, then, keeping in mind that the wind's vector is i:

r_p' * r_u' * r_g' * i * r_g * r_u * r_p = -i / sqrt(2) + j / 2 + k / 2.

And we

**know** this is right, because we have used

**math**, and

**math** is

**never wrong**!!

It's also intuitively about right: there's a good amount of wind against the plane's nose (-i), a good amount of wind blowing in the starboard direction across the plane (+j), and a good amount of wind down (+z), which is your "downward component".

There. Another problem in 3D geometry solved easily and plainly by use of simple arithmetic.

* Note: sane people*** need not read this post. However, if you

*are* sane and you take the trouble to wrap your brain around it, you can read it again later in comfort, secure in the knowledge that you won't have any remaining sanity to lose!

** heh heh

*** "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." — Lord Kelvin, 1892. (from

http://en.wikipedia.org/wiki/Quaternions#Quotes)