The term vector can be used in different ways. In the field of physics, a vector is a quantity defined by its point of application, its direction, its sense and its quantity.
Coplanar, in turn, is a concept that is not part of the dictionary of the Real Academia Española (RAE). On the other hand, the adjective coplanar appears, which refers to figures or lines that are in the same plane. In addition to the notion being incorrect according to the grammatical rules of our language, the idea of coplanar refers to points that are in the same plane (that is, they are coplanar points). When the point does not belong to that plane, it is considered not coplanar in relation to the others.
Coplanar vectors, therefore, are vectors that are in the same plane. To determine this question, the operation known as triple scalar product or mixed product is used. When the result of the triple dot product equals 0 , the vectors are coplanar (as are the points they join). In this sense, starting from the meaning and sense that coplanar vectors have, we can determine two remarkable statements that are worth taking into account:
-If there are only two vectors, they will always be coplanar.
-However, if you have more than two vectors, it may happen that one of them is not coplanar.
-Three vectors are coplanar or coplanar if their mixed product is zero.
-Three vectors can be considered coplanar or coplanar if they turn out to be linearly dependent. These guidelines also allow us to state that, when the result of the mentioned operation is different from 0, the vectors are non-coplanar. This means that these vectors, unlike coplanar vectors, are not part of the same plane. For example, vectors A (1, 1, 2) , B (1, 1, 1) and C (2, 2, 1) are coplanar vectors, since their triple dot product is 0 . In addition to this type of coplanar vectors, it should be taken into account that there are others that are also studied, such as:
-Simultaneous vectors, which are identified because their guidelines or lines of action intersect at a certain point.
-The parallel vectors, which are the vectors that are characterized because the lines that contain them are parallel.
-Sliding vectors, which have the particularity of being able to change their position along their orientation line.
– Position vectors. They are also known as fixed vectors and are identified because they have a fixed origin and because they come to record what a force in space is.
-Collinear vectors, which are identified because their lines of action are on the same line.
-Free vectors. They are those that have the ability to move in parallel lines or along their directions without being forced to undergo modifications of any kind.