Question

Coplanarity of Two Lines of three dimensional geometry.

Answer 1

Let two known lines are

\(\overrightarrow r = \overrightarrow {a_1} + \lambda \overrightarrow {b_1}\) ........(1)

and \(\overrightarrow r = \overrightarrow {a_2} + \lambda \overrightarrow {b_2}\) .....(2)

Line (1) passes through the point A, whose position vector is \(\overrightarrow {a_1}\) and is parallel to \(\overrightarrow {b_1}\)

Line (2) passes through the point B, whose position \(\overrightarrow {a_2}\) and is parallel to \(\overrightarrow {b_2}\)

\(\overrightarrow {AB} = \overrightarrow {a_2} - \overrightarrow {a_1}\)

The given lines will be coplanar if and only if \(\overrightarrow {AB} \) is perpendicular to \(\overrightarrow {b_1} \times \overrightarrow {b_2}.\)

Cartesian Form:

Let coordinates of points A and B are (x₁ y₁ z₁) and (x₂, y₂, z₂) respectively and direction ratios of \(\overrightarrow {b_1} \ and\ \overrightarrow {b_2}\) are a₁ b₁ c₁ and a₂, b₂, c₂ respectively.

Then \(\overrightarrow {AB} \) = (x₂ - x₁)î + (y₂ - y₁) ĵ + (z₂ - z₁) k̂

\(\overrightarrow {b_1}\) = a₁î + b₁ ĵ + c₁k̂

and \(\overrightarrow {b_1}\) = a₂î + b₂ ĵ + c₂k̂

The given lines are coplanar if and only if