How to Determine Whether Given 4 Points are Coplanar? For example, in the above figure, A, B, E, and F are non-coplanar points. But four or more points are non-coplanar if they don't lie on a plane. Two points are never non-coplanar and three points are also never non-coplanar. Non Coplanar Points Definition in Geometry If any 3 points are taken at a time, a plane can pass through all those 3 points, and hence they are coplanar.But each of F and E are NOT coplanar with A, B, C, and D.Here are some coplanar points examples from the above figure: Remember that given any two points are always coplanar and given any three points are always coplanar. So we define the coplanar points and non-coplanar points as follows with respect to the following example:įour or more points that lie on the same plane are known as coplanar points. But four or more points in 3D may not be coplanar. In the same way, three points in 3D can always pass through a plane and hence any 3 points are always coplanar. We know that two points in 2D can always pass through a line and hence any two points are collinear. Since, the scalar triple product is not equal to zero, hence the points A, B, C, and D are not coplanar.Īre four points A = (1, 5, 7), B = (6, 3, 1), C = (2 ,9, 5), and D = (7, 6, 5) coplanar? SolutionThe points that lie on the same plane are called coplanar points and hence the points that do NOT lie on the same plane are called non-coplanar points. Now, we will find the determinant of the above vectors like this: To find whether these points are coplanar or not, first, we will find, and like this: Now, we will start solving the questions in which four points will be given and we will check the coplanarity of those 4 points. So far, we have solved the problems in which we were given three points. Hence, we can conclude that the point A, B, and C are not coplanar because the scalar triple product of the three vectors is not equal to zero. Hence, we can conclude that the point A, B, and C are coplanar because the scalar triple product of the three vectors is equal to zero.ĭetermine if points A = (5, 1, 1), B = (3, 3, 1) and C = (2, 2, 1) are coplanar or not. Hence, we can conclude that the point A, B, and C are not coplanar because the scalar triple product of the three vectors is not zero.ĭetermine if points A = (0, 1, -1), B = (4, 3, 1) and C = (3, 2, 1) are coplanar or not. Now, we will calculate the dot product of and like this: We will use the formula for finding a determinant of 3 x 3 matrix to calculate the cross product of and. The elements of the determinant will be the coordinates of these vectors. The points A, B, and C will be coplanar if the scalar triple product of, , and is equal to zero.įirst, we will find the cross product of by using a determinant. In the next section, we will solve a couple of examples in which we will determine whether the given vectors are coplanar or not by using the scalar triple product.ĭetermine if point A = (5, 2, 3), B = (1, 6, 7) and C = (4, 2, 5) are coplanar or not. Mathematically, the scalar triple product is represented as: It is referred to as a scalar product because just like a dot product, the scalar triple product gives a single number. This product is equal to the dot product of the first vector by the cross product of other two vectors and. The scalar triple product of three vectors, , and can be mathematically denoted like this: The scalar triple product, also known as a mixed product, is the scalar product of three vectors.
To determine whether the three vectors are coplanar or not, we often find the scalar triple product of the three vectors.
If the scalar triple product of three vectors in 3D space is equal to zero, then we can say that these three vectors are coplanar.The scalar triple product of the coplanar vectors is always equal to zero. If two lines are coplanar in a three-dimensional space, then we can represent them in a vector form. We can easily find any two random coplanar vectors in a plane. These vectors are parallel to the same plane. The vectors that lie on the same plane in a three-dimensional space are referred to as coplanar vectors. In this article, we will discuss what are coplanar vectors with examples.