Question

Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).

Answer 1

The points of two lines are given as

A( 4,7,8 ),B( 2,3,4 )and C( −1,−2,1 ),D( 1,2,5 ) respectively.

The two lines with direction ratios ( a 1 , b 1 , c 1 ) and ( a 2 , b 2 , c 2 ) are said to be parallel when,

a 1 a 1 = b 1 b 2 = c 1 c 2

Also, the line passing through ( x 1 , y 1 , z 1 ),( x 2 , y 2 , z 2 ) has direction ratios, ( x 2 − x 1 ),( y 2 − y 1 ),( z 2 − z 1 ).

Then, direction ratios of given line AB are given as,

a 1 =2−4, b 1 =3−7, c 1 =4−8 a 1 =−2, b 1 =−4, c 1 =−4

And the direction ratios of line CD are given as,

a 2 =1−( −1 ), b 2 =2−( −2 ), c 2 =5−1 a 2 =2, b 2 =4, c 2 =4

Now,

a 1 a 2 = −2 2 =−1 b 1 b 2 = −4 4 =−1 c 1 c 2 = −4 4 =−1

So, a 1 a 1 = b 1 b 2 = c 1 c 2 =−1

Hence, the given two lines are parallel to each other.