tell me the answer of this question?

Question

show by the vector method that the conditions of the diagonals of a parralelogram to be mutually perpendicular are that the adjacent sides are equal in lenght

Answer 1

Actually, it's surprisingly simple. Any two non-collinear vectors define a parallelogram. If you're familiar with the parallelogram method of vector addition you know what I'm talking about. If you're not, you'll find the following page much more informative than any written description I could provide:

http://www.nhn.ou.edu/~walkup/demonstrations/WebTutorials/Parallelogram.htm

If its true that the parallelogram method strictly applies only to vector addition, it's easy to extend its usage for vector subtraction as well, since in order to subtract one vector from another you only have to invert the direction of the former vector, then perform the addition as usual. Although not strictly necessary, the proof will be easier to follow with the aid of a picture. Draw a pair of arbitrary vectors A, B; the angle between them can be anything you please, excepting 0°.

Complete the parallelogram by drawing parallel vectors to the original ones, just as in addition/subtraction by the parallelogram method. Having done this, it should become obvious that the diagonals of the parallelogram are vectors C = A + B and D = A − B. Now, C and D are mutually perpendicular if the dot product C · D = 0. That is,

(A + B) · (A − B) = A·A − A·B + A·B − B·B = A² − B² = 0.

Since both vectors have non-zero magnitude, it should be clear that this condition can only be met if |A| = |B|; that is, if their magnitudes are equal, which implies the sides of the parallelogram -the vectors themselves- are of equal length.

You can prove this graphically yourself, by drawing two equal-magnitude vectors, at any angle you choose, completing the parallelogram, then drawing the diagonals. You'll find that the angle between diagonals is 90°.

Answer 2

The diagonals of a parallelogram are only mutually perpendicular in the special case of a rectangle. The adjacent sides are not equal in length. Alas! It can not be shown by any method.
Use your spell checker in asking questions.

Answer 3

You create a 90 degree diagram of two vectors one verticle and one horizontal to show that both of the vertical elements and horizontal elements of the parralelogram diagonals are the same. You can't really write out the procedure here, since it is really a slope ratio of the diagonal sides

Answer 4

Since it is hard to draw pictures without an external web page, try this. Set up two candidate diagonals on the x and y axis intersecting at the origon. Arbitrarily call the length of the diagional on the y axis as 2. All the length of the diagional on the x axis a 2k. I think you can show that all lengths of the parallelogram are of length
sqrt(k^2+1), which is what you desire.

Answer 5

the adjacent sides are always equal ina parallelogram a.hole