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When are three vectors on the same plane?



Find a number s that garantees that the vectors i, i+j+k, and i+2j+sk will all be parallel to the same plane.


Three vectors are on a plane when and only when they generate no volume, i.e. their triple scalar product is 0.

Let u,v and w be the three given vectors,

> u := [1,0,0]: v := [1,1,1]: w := [1,2,s]:

their triple scalar product is,

> tsp := dotprod(u,crossprod(v,w));

                                 tsp := -2 + s

s is real (not complex) so its congugate is itself. Hence the value of s that makes tsp = 0 is

> s = 2:

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Carlos Rodriguez <>
Last modified: Thu Sep 14 15:16:19 EDT 2000