Determine Plane Containing 3 Points?

📌 CONCEPT: To determine whether three non-collinear points lie on a single plane, we need to check if the vectors formed by these points are coplanar, i.e., they lie in the same plane.

📐 RULE / FORMULA: If the scalar triple product of the position vectors of the three points is zero, then the points lie on a single plane. The scalar triple product is given by (a × b) . c = 0, where a, b, and c are vectors representing the position vectors of the three points.

💡 WORKED EXAMPLE: Let A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) be the three points. Then, the position vectors of A, B, and C are a = (1, 2, 3), b = (4, 5, 6), and c = (7, 8, 9) respectively. We can calculate the scalar triple product as follows: (a × b) . c = (−1, 2, 3) . (7, 8, 9) = −7 + 16 + 27 = 16 ≠ 0. Therefore, the points A, B, and C do not lie on a single plane.

⚠️ COMMON MISTAKE: Students often forget to calculate the scalar triple product correctly, which leads to incorrect results. They should ensure that the cross product (a × b) is calculated correctly before computing the dot product with c.