Two vectors A and B are added. Show that the magnitude of the resultant cannot be greater than A+B or smaller than l A-B l.
To show that the magnitude of the resultant vector cannot be greater than |A + B| or smaller than |A - B|, we'll use the properties of vector addition and the triangle inequality.
Let's start by considering the sum of vectors A and B, denoted as A + B. According to the triangle inequality, the magnitude of a sum of two vectors is always less than or equal to the sum of their individual magnitudes. In equation form, this can be expressed as:
|A + B| ≤ |A| + |B| ........ (1)
Similarly, if we subtract vector B from vector A, denoted as A - B, the magnitude of the resultant will be less than or equal to the sum of the individual magnitudes:
|A - B| ≤ |A| + |B| ........(2)
Now, let's consider the maximum possible magnitude of the resultant vector. It occurs when the vectors A and B are collinear and pointing in the same direction. In this case, the magnitude of the resultant is equal to the sum of the magnitudes of A and B. Mathematically, we can express this as:
|A + B| = |A| + |B| ........(3)
Combining equations (1) and (3), we get:
|A + B| ≤ |A + B|
This shows that the magnitude of the resultant vector, |A + B|, cannot be greater than the sum of the magnitudes of A and B.
Similarly, let's consider the minimum possible magnitude of the resultant vector. It occurs when the vectors A and B are collinear, but pointing in opposite directions. In this case, the magnitude of the resultant is equal to the difference between the magnitudes of A and B. Mathematically, we can express this as:
|A - B| = |A| - |B| ........(4)
Combining equations (2) and (4), we get:
|A - B| ≤ |A - B|
This shows that the magnitude of the resultant vector, |A - B|, cannot be smaller than the difference between the magnitudes of A and B.
In conclusion, the magnitude of the resultant vector cannot be greater than |A + B| or smaller than |A - B|.