Can two planes be parallel in more than one way?

📌 CONCEPT: Two planes in 3D space are parallel if their normal vectors are parallel, regardless of their orientation and position.

📐 RULE / FORMULA: The normal vectors of two parallel planes are scalar multiples of each other, i.e., there exists a constant k such that a = kb, where a and b are the normal vectors of the planes.

💡 WORKED EXAMPLE: If we have two planes with normal vectors a = (1, 0, 2) and b = (2, 0, 4), we can see that b is a scalar multiple of a (k = 2), confirming that the planes are parallel. Similarly, if we have a = (1, 0, 0) and b = (3, 0, 0), we can see that b is also a scalar multiple of a (k = 3), confirming that the planes are parallel.

⚠️ COMMON MISTAKE: Students often make the mistake of assuming that two planes are parallel if their normal vectors are proportional, without considering the possibility of different scalar multiples.