Question

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Answer 1

The given points are A( 2,3,4 ), B( −1,−2,1 ) and C( 5,8,7 ).

The general formula for the direction ratio of the line joining the two points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is x 2 − x 1 , y 2 − y 1 and z 2 − z 1 .

So, the direction ratios of AB are given as,

A( x 1 , y 1 , z 1 )=( 2,3,4 ) B( x 2 , y 2 , z 2 )=( −1,−2,1 ) ( x 2 − x 1 ),( y 2 − y 1 ),( z 2 − z 1 )=( −1−2 ),( −2−3 ),( 1−4 ) =−3,−5,−3

The direction ratios of BC are given as,

B( x 1 , y 1 , z 1 )=( −1,−2,1 ) C( x 2 , y 2 , z 2 )=( 5,8,7 ) ( x 2 − x 1 ),( y 2 − y 1 ),( z 2 − z 1 )=( 5−( −1 ) ),( 8−( −2 ) ),( 7−1 ) =6,10,6

Direction ratios of BCare −2 times the direction ratios of AB, it means both are proportional. ABis parallel to BCsince Bis common to both ABand BC.

Hence, points A, Band Care collinear.