(ii) Two co-planar straight lines are either parallel or they intersect at a point. (i) Lines or points are said to be co-planar if they lie on the same plane in other words, lines or points are co-planar if a plane can be made to pass through them. The statement that a straight line lies wholly on a surface signifies that every point on the line (however produced in both directions) lies on the surface.Ī surface is called curved surface when it is not a plane surface. Similarly, planes are also assumed to be of infinite extent, unless otherwise stated. (vii) Plane or Plane Surface: If the straight line joining two points on a surface lies wholly on the surface then the surface is called a plane surface or a plane.Ī straight line may be extended indefinitely in either direction, that is, straight lines are supposed to be of infinite length. (vi) Solid Geometry: The branch of geometry which deals with the properties of points, lines, surfaces and solids in three dimensional space is called solid geometry. In other words, a line is generated by the motion of a point, a surface is generated by the motion of a line and a solid is generated by the motion of a surface. 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 book is a solid, each of its six faces is a surface, each of its edges is a line and each of its corners is a point.Ī line is bounded by points, a surface is bounded by lines and a solid is bounded by surfaces. 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.
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.
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