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Question

Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

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Solution

It is given that the line passing through the points ( 1,1,2 ) and ( 3,4,2 ) is perpendicular to the line passing through the points ( 0,3,2 )and ( 3,5,6 ).

The two lines have direction ratios a,b,cand a , b , c .

The condition for perpendicularity is,

a a +b b +c c =0(1)

Given, a line passes through points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ), then the direction cosine is,

( x 2 x 1 ),( y 2 y 1 ),( z 2 z 1 )

The direction ratios of given lines are,

a=( 31 ),b=4( 1 ),c=22 a=2,b=5,c=4

And,

a =( 30 ), b =( 53 ), c =( 62 ) a =3, b =2, c =4

Substitute these values in equation (1).

a a +b b +c c =( 2×3 )+( 5×2 )+( 4×4 ) =6+10+( 16 ) =1616 =0

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


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