Understanding Coplanar Points: Exploring the Concept of Points in the Same Plane

Non-coplanar

In mathematics, the term “coplanar” refers to a set of points or vectors that lie on the same plane

In mathematics, the term “coplanar” refers to a set of points or vectors that lie on the same plane. If a set of points or vectors is not coplanar, then they do not lie on the same plane.

To better understand this concept, let’s consider three points in three-dimensional space: A, B, and C.

If these three points A, B, and C lie on the same plane, they are considered coplanar. This means that a flat surface can be formed that contains all three points.

However, if these points are not coplanar, it means that they do not lie on the same plane. In other words, no matter how you try to position them, you cannot form a single flat plane that contains all three points.

To illustrate this further, let’s consider an example. Suppose we have three points in three-dimensional space: A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9).

To check if these points are coplanar or not, we can calculate the vectors formed by connecting any two points. We can find the vectors AB and AC as follows:

Vector AB = B – A = (4, 5, 6) – (1, 2, 3) = (3, 3, 3)
Vector AC = C – A = (7, 8, 9) – (1, 2, 3) = (6, 6, 6)

Now, if AB and AC are parallel or linearly dependent, it means that the three points are coplanar. However, if they are not parallel or linearly independent, it means that the three points are non-coplanar.

To check if vectors AB and AC are parallel, we can calculate their cross product. If the cross product is the zero vector, it means that the vectors are parallel and, consequently, the three points are coplanar.

Calculating the cross product of AB and AC, we have:

AB x AC = (3, 3, 3) x (6, 6, 6) = (0, 0, 0)

Since the cross product is the zero vector, it means that vectors AB and AC are parallel. Therefore, the three points A, B, and C are coplanar.

In summary, if three points (or vectors) lie on the same plane, they are coplanar. However, if they do not lie on the same plane, they are non-coplanar. This concept is important in various areas of mathematics, such as geometry, linear algebra, and vector analysis.

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