What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6

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Question: What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)

Options:

Correct Answer: 0 = 0

Exam Year: 2021

Solution:

The points are collinear, hence the equation of the plane is 0 = 0.

What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)

What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)

Step 1: Understand that a plane in 3D space can be defined by three non-collinear points.
Step 2: Identify the given points: (1, 2, 3), (4, 5, 6), and (7, 8, 9).
Step 3: Check if the points are collinear by finding the vectors between them.
Step 4: Calculate the vector from the first point to the second point: (4-1, 5-2, 6-3) = (3, 3, 3).
Step 5: Calculate the vector from the first point to the third point: (7-1, 8-2, 9-3) = (6, 6, 6).
Step 6: Check if the second vector is a scalar multiple of the first vector. Here, (6, 6, 6) = 2 * (3, 3, 3).
Step 7: Since the second vector is a scalar multiple of the first, the points are collinear.
Step 8: Conclude that since the points are collinear, they do not define a unique plane.
Step 9: Therefore, the equation of the plane is 0 = 0, indicating that any plane can pass through these points.

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