Find the equations of the two lines through the origin which intersect the line 2 x − 3 = 1 y − 3 = 1 z at an angle of 3 π .

Question

Find the equations of the two lines through the origin which intersect the line 2 x − 3 ​ = 1 y − 3 ​ = 1 z ​ at an angle of 3 π ​ .

TestHub · Accepted Answer

Parametrize the point P on the given line. The line OP must make an angle of 60 degrees with its direction vector d: P = ( 3 + 2 t , 3 + t , t ) , d = ( 2 , 1 , 1 ) , d ⋅ d = 6. 4 ( P ⋅ d ) 2 = ( P ⋅ P ) ( d ⋅ d ) , 4 ( 6 t + 9 ) 2 = 6 ( 6 t 2 + 18 t + 18 ) , 108 ( t + 1 ) ( t + 2 ) = 0. t = − 1 : P = ( 1 , 2 , − 1 ) , t = − 2 : P = ( − 1 , 1 , − 2 ) . r = s ( 1 , 2 , − 1 ) or r = s ( − 1 , 1 , − 2 ) . ​

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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