you need

- - paper;
- - handle;
- - line.

Guide

1

vector - a directed segment, that is, a value considered to be given in full, if you set the length and direction (angle) to a given axis.The position of the vector no longer limited.Still considered two vectors have the same length and a direction.Therefore, when using the coordinates of the vectors represent the radius vectors of the points of its end (beginning is at the origin).

2 By definition: geometric resultant vector is the vector sum of the vectors emanating from the first end and having a second end, with the proviso that first end, aligned with the beginning of the second.It goes on and on, building a chain of similarly situated vectors.

Draw Quadrilateral ABCD is defined by the
vectors a, b, c, and d according to Fig.1. It is obvious that with this arrangement, the resultant vector d = a + b + c.

Draw Quadrilateral ABCD is defined by the

3

scalar product in this case, it's best to determine on the basis vectors a, and d.The scalar product denoted by (a, d) = | a || d | cosf1.Where F1 - the angle between a and d.

scalar product of vectors, given the coordinates, determined by the following expression:

(a (ax, ay), d (dx, dy)) = axdx + aydy, | a | ^ 2 = ax ^ 2 + ay ^ 2, | d |^ 2 = dx ^ 2 + dy ^ 2, then

cos F1 = (axdx + aydy) / (sqrt (ax ^ 2 + ay ^ 2) sqrt (dx ^ 2 + dy ^ 2)).

scalar product of vectors, given the coordinates, determined by the following expression:

(a (ax, ay), d (dx, dy)) = axdx + aydy, | a | ^ 2 = ax ^ 2 + ay ^ 2, | d |^ 2 = dx ^ 2 + dy ^ 2, then

cos F1 = (axdx + aydy) / (sqrt (ax ^ 2 + ay ^ 2) sqrt (dx ^ 2 + dy ^ 2)).

4

Basic concepts of vector algebra in relation to the task at hand, lead to the fact that the unique setting of this task quite a job three vectors located, say, AB, BC, and CD, you have a,b, c.You can of course just set the coordinates of points A, B, C, D, but this method is redundant (option 4 instead of 3).

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example.Quadrilateral ABCD is given by the vector of its sides AB, BC, CD a (1,0), b (1,1), c (-1,2).Find the angle between the parties.

decision.In connection with the above, the 4-th vector (for AD)

d (dx, dy) = a + b + c = {ax + bx + cx, ay + by + cy} = {1,3}.Following the procedure of calculating the angle between the vectors and

cosf1 = (axdx + aydy) / (sqrt (ax ^ 2 + ay ^ 2) sqrt (dx ^ 2 + dy ^ 2)) = 1 / sqrt (10), F1 = arcos (1 / sqrt (10)).

-cosf2 = (axbx + ayby) / (sqrt (ax ^ 2 + ay ^ 2) sqrt (bx ^ 2 + by ^ 2)) = 1 / sqrt2, p2 = arcos (-1 / sqrt2), p2 = 3p/4.

-cosf3 = (bxcx + bycy) / (sqrt (bx ^ 2 + by ^ 2) sqrt (cx ^ 2 + cy ^ 2)) = 1 / (sqrt2sqrt5), F3 = arcos (-1 / sqrt (10)) = n-F1.

In accordance with Remark 2 - F4 = 2n F1 - F3 f2- = n / 4.

decision.In connection with the above, the 4-th vector (for AD)

d (dx, dy) = a + b + c = {ax + bx + cx, ay + by + cy} = {1,3}.Following the procedure of calculating the angle between the vectors and

cosf1 = (axdx + aydy) / (sqrt (ax ^ 2 + ay ^ 2) sqrt (dx ^ 2 + dy ^ 2)) = 1 / sqrt (10), F1 = arcos (1 / sqrt (10)).

-cosf2 = (axbx + ayby) / (sqrt (ax ^ 2 + ay ^ 2) sqrt (bx ^ 2 + by ^ 2)) = 1 / sqrt2, p2 = arcos (-1 / sqrt2), p2 = 3p/4.

-cosf3 = (bxcx + bycy) / (sqrt (bx ^ 2 + by ^ 2) sqrt (cx ^ 2 + cy ^ 2)) = 1 / (sqrt2sqrt5), F3 = arcos (-1 / sqrt (10)) = n-F1.

In accordance with Remark 2 - F4 = 2n F1 - F3 f2- = n / 4.

Note

Note 1. In the definition of the inner product used angle between the vectors.Here, for example, Q2 - is the angle between AB and BC, and between a and b, this angle p-p2.cos (n * 2) = - sosf2.Similarly, for the F3.

Note 2. It is known that the sum of the angles of a quadrilateral is equal to 2n.Therefore, F4 = 2n F1 - F3 f2-.

Note 2. It is known that the sum of the angles of a quadrilateral is equal to 2n.Therefore, F4 = 2n F1 - F3 f2-.