| Brett
Thompson |
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 |
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| David
Loeffler |
I think w is meant to be [wx , wy , wz ], rather than a vector proportional to [x,y,z]. David |
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| Sarah
Sarah |
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 |
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| Edwin
Chan |
Brett, here's a ''simplification'' if you can call it that: Direction of w×(w×r) w×(w×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 q is the angle between w and r then (p- q) is the angle between w and (w×r) Magnitude of w×(w×r) The magnitude of (w×r) is |w||r|sinq where |w| and |r| are the respective magnitudes of w and r Therefore the magnitude of w×(w×r) is |w||(w×r)|sin(p-q) =|w||w||r|sinqsin(p-q) =|w|2|r|sinq |
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| Edwin
Chan |
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. |