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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=( 3−1 ),b=4−( −1 ),c=−2−2 a=2,b=5,c=−4 And, a ′ =( 3−0 ), b ′ =( 5−3 ), c ′ =( 6−2 ) 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 ) =16−16 =0 Hence, the given two lines are perpendicular to each other.

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