Question

Show that the three lines with direction cosines 12/13,-3/13,-4/13,4/13,12/13,3/13,3/13,-4/13,12/13 are mutually perpendicular.

Answer 1

Answer:

three lines are mutually perpendicular

Step-by-step explanation:

12/13,-3/13,-4/13

4/13,12/13,3/13

3/13,-4/13,12/13

Two lines with ditection cosines

L₁ , M₁ , N₁  & L₂ , M₂ , N₂ are perpendicular if

L₁L₂  + M₁M₂ + N₁N₂  = 0

taking 1st two lines

L₁ = 12/13  , M₁ = -3/13 , N₁ = -4/13

L₂ = 4/13  , M₂ = 12/13 , N₂ = 3/13

L₁L₂  + M₁M₂ + N₁N₂

= (48/169 - 36/169 - 12/169)

= 0

=> perpendicular

taking 1st & 3rd Line

L₁ = 4/13  , M₁ = 12/13 , N₁ = 3/13

L₂ = 3/13  , M₂ = -4/13 , N₂ = 12/13

L₁L₂  + M₁M₂ + N₁N₂

= ( 12/169 -48/169 + 36/169)

= 0

=> perpendicular

Taking 2nd & 3rd Line

L₁ = 12/13  , M₁ = -3/13 , N₁ = -4/13

L₂ = 3/13  , M₂ = -4/13 , N₂ = 12/13

L₁L₂  + M₁M₂ + N₁N₂

= ( 36/169 + 12/169 - 48/169)

= 0

=> perpendicular

three lines are mutually perpendicular