Collinearity Of Three Points
Decide whether the three points (1, 2, 3), (2, 4, 6), and (4, 8, 12) are collinear in three-dimensional space.
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About This Question
- Subject
- mathematics
- Chapter
- three dimensional geometry
- Topic
- collinearity of three points
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
They are collinear
The criterion is that three points are collinear when the vectors joining them are parallel, that is one is a scalar multiple of the other, equivalently their cross product is the zero vector. This parallel-vector test is a clean JEE Advanced standard. Take A = (1, 2, 3), B = (2, 4, 6), C = (4, 8, 12). Then AB = B - A = (1, 2, 3) and AC = C - A = (3, 6, 9) = 3(1, 2, 3) = 3 \cdot AB. Since AC is a scalar multiple of AB, the vectors are parallel and share point A, so the three points lie on one line. Equivalently AB \times AC = (1,2,3) \times (3,6,9) = (2\cdot9 - 3\cdot6,\ 3\cdot3 - 1\cdot9,\ 1\cdot6 - 2\cdot3) = (0, 0, 0). The triangle option fails because zero cross product means zero area. The coplanar-not-collinear option is wrong since proportionality is stronger than coplanarity. No two points coincide, as all coordinates differ. This applies the collinearity-via-parallel-vectors theorem. Plausibility check: a vanishing cross product confirms zero enclosed area, the hallmark of collinear points.
This easy difficulty mathematics question is from the chapter three dimensional geometry, covering the topic of collinearity of three points. It appeared in the 2025 exam.
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