Finding Equation of Plane?

📌 CONCEPT: The equation of a plane passing through three non-collinear points can be found using the normal form of the equation of a plane, which involves the cross product of two vectors formed by the points.

📐 RULE / FORMULA: The equation of the plane passing through points A, B, and C can be found using the formula:

r⋅n = d, where r is the position vector of any point on the plane, n is the unit normal vector to the plane, and d is the distance of the plane from the origin.

💡 WORKED EXAMPLE: Let A(2, 3, 1), B(1, 2, 2), and C(3, 1, 3) be the given points. First, we find the vectors AB and AC: AB = (-1, -1, 1) and AC = (1, -2, 2). Then, we find the cross product of these vectors to get the normal vector to the plane: n = (-3, 0, 1). Next, we find the unit vector of n: n = (-3/√10, 0, 1/√10). Now, we find the equation of the plane using the formula: r⋅n = d.

⚠️ COMMON MISTAKE: Students often get the direction ratios of the normal vector to the plane incorrect, which leads to an incorrect equation of the plane. Also, they might forget to normalize the normal vector, resulting in an incorrect equation of the plane.