Angle between Two Intersecting Lines

Let v = a₁ + λb₁ and v = a₂ + λb₂ be the equations of two straight lines. If θ is the angle between them, then cos θ = (b₁ - b₂)/ |b₁| |b₂|

Also if θ is the angle between (x - x₁)/ a₁ = (y - y₁)/ b₁ = (z - z₁)/ c₁

And (x - x₁)/ a₂ = (y - y₁)/ b₂ = (z - z₁)/ c₂

Then cos θ = (a₁a₂ + b₁b₂ + c₁c₂)/ √ (a₁² + b₁² + c₁²) √ (a₂² + b₂² + c₂²)

• Condition of perpendicularity:

The lines are perpendicular, if b₁ - b₂ = 0 or a₁a₂ + b₁b₂ + c₁c₂ = 0

• Condition of parallelism:

The lines are parallel, if b₁ = λb₂ for some scalar λ or a₁/a₂ = b₁/b₂ = c₁/c₂

Example:

Find the angle between the pair of lines

v = 3i + 2j – 4k + λ (I + 2j + 2k)

And v = 5i — 2k + µ (3i + 2j + 6k)

Solution:

Let the angle is θ then

cos θ = (b₁ - b₂)/ |b₁| |b₂|

= (i + 2j + 2k) (3i + 2j – 6k)/√ (1 + 4 + 4) √ (9 + 4 + 36)

= (3 + 4 + 12)/ √9 √49

= 19/ 21

θ = cos⁻¹ (19/21)