Answer to Question #93795 in Mechanics | Relativity for Shirshak Aryal

Question

Show that the magnitude of the resultant of two vectors A and B cannot be greater than the sum of the magnitudes of A and B, and lesser than the difference of the magnitudes of A and B.

Answer 1

The sum of 2 vectors is defined in following way.

Using parallel transfer, combine the beginning of one vector with the end of another. Then, the sum of these 2 vectors is the vector, which starts at the beginning of one vector and ends at the end of another. (please, see picture)

Thus, beginning and ends of two vectros and of their sum form a triangle or a line. Magnitudes of these 2 vectors and of their sum are lengths of respective sides of the triangle.

If they form a triangle, then, according to triangle inequlity, the lenghth of one side of a triangle is lesser than sum of 2 others. If they form a line, then magditude of the sum is equal to the sum of magnitudes or to the difference of magnitudes. If it is equal to the difference of magnitudes,than it is lesser than their sum,because magnitudes are positive numbers. Therefore, the magnitude of the sum is lesser or equal to sum of magnitudes, or, in other words, not greater than the sum of magnitudes.

Now let's look at the difference of magnitudes. Beginning and ends of two vectros and of their sum form a triangle or a line. If they form a triangle, then according to triangle inequlity, the lenghth of one side of a triangle is bigger than difference of 2 others. If they form a line, then magditude of the sum is equal to the sum of magnitudes or to the difference of magnitudes. If it is equal to the sum of magnitudes,than it is greater than their difference,,because magnitudes are positive numbers. Therefore, the magnitude of the sum is greater or equal to difference of magnitudes, or, in other words, not lesser than the difference of magnitudes.