Methods to prove collinearity:

• The 2 vectors    are collinear if there exists kÎR such that      .
• If   then    are collinear.
• The 3 points   are collinear if there exists k ∈ R such that  
•  .
• If   then A, B, C are collinear.
•    are collinear if there exists scalars l, m, n, such that    where l + m + n = 0.

All above methods are equivalent and any of them can be utilized to prove collinearity.

 Illustration: Let  be 3 non-zero vectors such that any 2 of them are non-collinear. If +2 is collinear to  and +3 is collinear with, then prove that.

Key concept:           Two vectors  are collinear if there exists k∈R such that .

Solution:                   It is given that  +2 is collinear with 

                                    for some scalar λ                  ...(i)

                                    Also+3 is collinear with 

                                    =>+3= μa for some scalarμ                 ...(ii)

                                    from (i) and (ii)

                                    Substituting the values of l and m in (i) & (ii), we get

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