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Calc3.2.CrossProductProof (view source)
Revision as of 05:42, February 28, 2010
, 05:42, February 28, 2010→Third Identity
[[Image:Cross product plane.jpg|right|200px|thumb|We imagine <math>\vec{p}</math> sticking out of the screen towards the reader. You may need to enlarge this picture.]]
[[Image:Cross product plane.jpg|right|200px|thumb|We imagine <math>\vec{p}</math> sticking out of the screen towards the reader. You may need to enlarge this picture.]]
So, let us assume that <math>\vec{q}_*, \vec{r}_*</math> both have non-zero magnitude and are not collinear. If we imagine <math>\vec{p}</math> sticking out of our computer monitors, pointing towards the reader, then the vectors in questions all look something like the picture on the right.
So, let us assume that <math>\vec{q}_*, \vec{r}_*</math> both have non-zero magnitude and are not collinear. If we imagine <math>\vec{p}</math> sticking out of our computer monitors, pointing towards the reader, then the vectors in questions all look something like the picture on the right.
If <math>\vec{s}</math> is any vector in the plane of your screen (ie, <math>p_1x+p_2y+p_3z=0</math>), then <math>\vec{p}\times\vec{s}</math> is ''also'' in the plane of your screen. It will make a <math>90^\circ=\pi/2</math> angle with <math>\vec{s}</math> and have magnitude <math>\left\|{ \vec{p} }\right\| \left\|{ \vec{s} }\right\|</math>, since <math>\sin(90^\circ)=1</math>. This gives rise to the picture at right. Letting <math>\vec{s}</math> be each of <math>\vec{q}_*, \vec{r}_*, \vec{q}_*+\vec{r}_*</math>, we see that the vectors <math>\vec{p}\times\vec{q}_*, \vec{p}\times\vec{r}_*, \vec{p}\times(\vec{q}_*+\vec{r}_*)</math> have the same configuration as <math>\vec{q}_*, \vec{r}_*, \vec{q}_*+\vec{r}_*</math>, just rotated <math>90^\circ</math> and multiplied by <math>\left\|{ \vec{p} \ }\right\|</math>. Convince yourself that this picture demonstrates <math>\vec{p}\times(\vec{q}_*+\vec{r}_*) = \vec{p}\times\vec{q}_* + \vec{p}\times\vec{r}_* \ </math>.
===Putting it all together===
==Notes==
==Notes==
{{reflist}}
{{reflist}}