Representation and Reference Systems
The simplest representation of a vector is as an arrow connecting two points. Thus,
Knowledge of the components of a vector enables one to compute its magnitude—in this case, 5, from the Pythagorean theorem [(32 + 42)1/2 = 5)]—and its direction from trigonometry, once the lengths of the sides of the right triangle formed by the vector and its components are known. (Trigonometry can also be used to find the component of the vector as projected in some direction other than the x -axis or y -axis.) Since the vector points from A to B, both its components are positive; if it pointed from B to A, its components would be [ - 3, - 4] but its magnitude and orientation would be the same.
It is obvious that an infinite number of vectors can have the same components [3,4], since there are an infinite number of pairs of points in the plane with x - and y -coordinates whose respective differences are 3 and 4. All these vectors have the same magnitude and direction, being parallel to one another, and are considered equal. Thus, any vector with components a and b can be considered as equal to the vector [ a, b ] directed from the origin (0,0) to the point ( a, b ). The concept of a vector can be extended to three or more dimensions.
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