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Question

Show that the vectore a, b and c are coplanar, if a+b, b+c and c+a are coplanar.

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Solution

Given
a,b,c are coplanar
a(b×c)=0
[abc]=0----------(1)
To show
(a+b)((b+c)×(c+a))=0
(a+b)(b×c+b×a+c×c+c×a)=0
a(b×c)+a(b×a)+a(c×c)+a(c×a)+b(b×c)+b(b×a)+b(c×c)+b(c×a)=0
[abc]+[aba]+[acc]+[aca]+[bbc]+[bba]+[bcc]+[bca]=0
but we know by definition of scalar triple product if two vectors is same in the triple scalar product then it is equal to zero
[aab]=0
SO
[abc]+0+0+0+0+0+0+[bca]=0
2[abc]=0
from eq (1)
0=0
Hence proved

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