Direction Ratios from a Plane and a Line?

📌 CONCEPT: Direction ratios of a line are the ratios of the x, y, and z coordinates of any point on the line to the respective direction ratios of the line.

📐 RULE / FORMULA: For a line to lie in a plane, the direction ratios of the line must be proportional to the direction ratios of the normal vector of the plane.

💡 WORKED EXAMPLE: Given a plane with normal vector <2, 3, -1> and a line with direction ratios <1, 2, 3>, we need to show that the line lies in the plane. Let's consider a point on the line (0, 0, 2). The direction ratios of the line are proportional to the normal vector of the plane, so the line lies in the plane.

⚠️ COMMON MISTAKE: Students may incorrectly assume that the line lies in the plane just because it passes through a point on the plane, without checking if the direction ratios are proportional to the normal vector.