Find the vector e

Given – The lines have direction ratios of (1, - 1, - 2) and (- 1, 0, 2). The plane parallel to these lines passes through (1, 2, 3)

To find – The vector 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

& , where the two vectors represent the directions

The equation of the plane maybe represented as - 2x - z + d = 0

Now, this plane passes through the point (1, 2, 3)

Hence,

(- 2) × 1 - 3 + d = 0

⇒ d = 5

The Cartesian equation of the plane : - 2x - z + 5 = 0 i.e. 2x + z = 5

The vector equation :