Show that the vectors a,b,c are coplanar if and only if a.b cross c =0?
回答 (3)
2013-04-25 12:07 pm
✔ 最佳答案
If a.(bxc) = 0 then vector a is perpendicular to vector bxc.But vector bxc is perpendicular to vectors b and c and
so a,b and c are coplanar.
If a,b,c are coplanar then a=pb+qc where p and q are scalars.
then a.(bxc) = pb(.bxc)+qc.(bxc)
but b.(bxc)=0 and c.(bxc)=0 so
a.(bxc0=0
2013-04-25 12:16 pm
assume a,b,c, not 0.
=> (a.b) x c = 0 means that a.b is not 0 and it defines a plane. Call this vector, z, where z = a.b in the plane. z x c = 0 for any c means that c and z are in the same plane by definition
<= a,b,c coplanar means that a.b = z is a coplanar vector also, z x c = 0, for all co planar vectors z and c.
=> (a.b) x c = 0 means that a.b is not 0 and it defines a plane. Call this vector, z, where z = a.b in the plane. z x c = 0 for any c means that c and z are in the same plane by definition
<= a,b,c coplanar means that a.b = z is a coplanar vector also, z x c = 0, for all co planar vectors z and c.
2013-04-25 11:56 am
a.(b X c)=0
Now, b and c are always coplaner. So, (b X c) is perpendicular with b and c)
again a.(b X c)=0, so (b X c) is perpendicular with a
So, (b X c) is perpendicular with all b,c and a.
That results in the fact that, a , b and c are perpendicular.
Now, b and c are always coplaner. So, (b X c) is perpendicular with b and c)
again a.(b X c)=0, so (b X c) is perpendicular with a
So, (b X c) is perpendicular with all b,c and a.
That results in the fact that, a , b and c are perpendicular.
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