Brett Thompson

Posted on Thursday, 26 June, 2003 - 07:21 am:

Alright I have the following problem.

w X (w X r)

X is a cross product.
w = [wx, wy, wz] (A 3D line in space)
r = [rx, ry, rz] (A 3D line in space)

I can get this down to the following
w(w.r)-r(w.w) (.= dot product)

Can someone show me how to simplify this further??

Cheers

David Loeffler

Posted on Thursday, 26 June, 2003 - 11:54 am:

I think w is meant to be [w_x , w_y , w_z ], rather than a vector proportional to [x,y,z].

David

Sarah Sarah

Posted on Thursday, 26 June, 2003 - 01:39 pm:

Are you sure it's possible to simplify? The geometrical interpretation of wx(wxr) is more obvious than that of any equivalent expression I can come up with - it's a vector in the same plane as r and w, perpendicular to w, with magnitude IwI² IrIsinq where q is the angle between w and r. Or, you could say, -IwI² times the projection of r perpendicular to w. Probably missed something, I usually do.. Does it help to draw some pictures?
Love Sarah

Edwin Chan

Posted on Saturday, 28 June, 2003 - 02:43 am:

Brett, here's a ''simplification'' if you can call it that:

Direction of $w \times (w \times r)$

$w \times (w \times r)$ turns out to be a vector in the plane of $w$ and $r$ but antiparallel to $r$ (pointing in the opposite direction) you can figure this out by repeated application of the right-hand rule. Furthermore note that if $θ$ is the angle between $w$ and $r$ then $(π - θ)$ is the angle between $w$ and $(w \times r)$

Magnitude of $w \times (w \times r)$

The magnitude of $(w \times r)$ is $| w | | r | \sin θ$ where $| w |$ and $| r |$ are the respective magnitudes of $w$ and $r$

Therefore the magnitude of $w \times (w \times r)$ is

$| w | | (w \times r) | \sin (π - θ)$

$= | w | | w | | r | \sin θ \sin (π - θ)$

$= | w |^{2} | r | \sin θ$

Edwin Chan

Posted on Saturday, 28 June, 2003 - 03:08 am:

Although Sarah's formula is slightly different from mine, I guess what we're both saying is that sometimes it makes for better understanding to consider direction & magnitude separately than to just apply a rote algebraic formula.