Given - & . A plane is parallel to both these lines and passes through (1, 2, - 4).

To find – The equation of the plane

Tip – A plane parallel to two vectors will have its normal in a direction perpendicular to both the vectors, which can be evaluated by taking their cross product

The direction ratios of the given lines are (2, 3, 6) and (1, 1, - 1)

&

The equation of the plane maybe represented as - 9x + 8y - z + d = 0

Now, this plane passes through the point (1, 2, - 4)

Hence,

(- 9) × 1 + 8 × 2 - (- 4) + d = 0

⇒ d = - 11

The Cartesian equation of the plane : - 9x + 8y - z - 11 = 0 i.e. 9x - 8y + z + 11 = 0

The vector equation :