Answer :

Given -


A = (2,1,3)


B = (5,0,5)


C = (-4,3,-1)


To prove – A, B and C are collinear


Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by ((a’-a),(b’-b),(c’-c))


The direction ratios of the line AB can be given by


((5-2),(0-1),(5-3))


=(3,-1,-2)


Similarly, the direction ratios of the line BC can be given by


((-4-5),(3-0),(-1-5))


=(-9,3,-6)


Tip – If it is shown that direction ratios of AB=λ times that of BC , where λ is any arbitrary constant, then the condition is sufficient to conclude that points A, B and C will be collinear.


So, d.r. of AB


=(3,-1,-2)


=(-1/3)Χ(-9,3,-6)


=(-1/3)Хd.r. of BC


Hence, A, B and C are collinear


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