Vector — a piece which has not only length, but also the direction. Vectors play a large role in mathematics, but especially in physics as the physics very often deals with sizes which are convenient for presenting in the form of vectors. Therefore in mathematical and physical calculations there can be a need to calculate length of the vector set by coordinates.
Instruction
1. In any system of coordinates the vector is defined through two points — the beginning and the end. For example, in the Cartesian coordinates on the plane the vector is designated as (x1, y1; x2, y2). In space, respectively, each point will have about three coordinates, and the vector will appear in a look (x1, y1, z1; x2, y2, z2). Certainly, the vector can be defined both for four-dimensional, and for any other space. It will be much more difficult to be presented, but in terms of mathematics all related calculations will remain the same.
2. Vector length still call it the module. If A is a vector, then |A| — the number equal to its module. For example, any real number can be presented as a one-dimensional vector from the beginning in zero point. Let's tell, number-2 will be a vector (0;-2). The module of such vector will be equal to a square root from a square of coordinate of its end, that is √ ((-2) ^2) = 2. In a general view, if A = (0, x), that |A| = √ (x^2). From this, in particular, follows that the module of a vector does not depend on its direction — numbers 2 and-2 are equal on the module.
3. Let's pass to the Cartesian coordinates to the planes. And in this case it is the simplest to calculate vector length if its beginning coincides with the beginning of coordinates. The square root will need to be taken from the sum of squares of coordinates of the termination of a vector. | 0, 0; x, y | = √ (x^2 + y^2). For example, if we have A vector = (0, 0; 3, 4), its |A| module = √ (3^2 + 4^2) = 5. Actually, you calculate the module on Pythagoras's formula about a hypotenuse of a rectangular triangle. The coordinate pieces setting a vector play a role of legs, and the vector serves as a hypotenuse which square, as we know, is equal to the sum of their squares.
4. When the beginning of a vector is not in a reference point of coordinates, calculation of the module becomes a little more labor-consuming. It is necessary to square not coordinates of the end of a vector, but the difference between the coordinate of the end and the corresponding coordinate of the beginning. It is easy to notice that if the coordinate of the beginning is equal to zero, then the formula turns into previous. You likewise use Pythagorean theorem here — the differences of coordinates become lengths of legs. If A = (x1, y1; x2, y2), that |A| = √ ((x2 - x1) ^2 + (y2-y1) ^2). Let's assume that to us A vector = is set (1, 2; 4, 6). Then its module is equal to |A| = √ ((4 - 1) ^2 + (6 - 2) ^2) = 5. If you construct this vector on the coordinate plane and compare it to previous, then you will easily see that they are equal among themselves, as becomes obvious at calculation of their length.
5. This formula is universal, and it is easy to generalize it on a case when the vector is located not on the plane, and in space, or even has more than three coordinates. Its length will be still equal to a square root from the sum of squares of differences of coordinates of the end and the beginning.