Can a plane pass through the intersection of two lines?

📌 CONCEPT: A plane passing through the intersection of two lines can indeed be represented in terms of a vector equation, but it requires specific conditions to be satisfied.

📐 RULE / FORMULA: The condition for a plane to pass through the intersection of two lines is that the normal vector to the plane should be perpendicular to both direction vectors of the lines.

💡 WORKED EXAMPLE: Consider two lines represented by vectors u = 2i + 3j + k and v = i + 2j + 2k. If we find a normal vector n = 2i + 3j - 4k that is perpendicular to both u and v, we can represent the plane passing through their intersection using the equation (a(u) + b(v)) ⋅ n = 0.

⚠️ COMMON MISTAKE: Students often overlook the fact that the normal vector to the plane should be perpendicular to both direction vectors of the lines, leading to incorrect representations of the plane.