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).
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 , c and 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.
It is given that the line passing through the points
The two lines have direction ratios
The condition for perpendicularity is,
Given, a line passes through points
The direction ratios of given lines are,
And,
Substitute these values in equation (1).
Hence, the given two lines are perpendicular to each other.
