can the magnitude of the resultant of two vectors be smaller than the magnitude of either of them? Explain
Yes, of course. As en extreme example, the resultant can be zero if the individual vectors have equal magnitudes in opposite directions.
No, the magnitude of the resultant of two vectors cannot be smaller than the magnitude of either of them. To understand why, let's consider vector addition geometrically. When two vectors are added together, their magnitudes and directions are taken into account.
The magnitude of a vector represents its "length" or the size of its physical quantity. If the magnitude of the resultant vector is smaller than one of the original vectors, it means that the two vectors would somehow cancel each other out partially or completely.
However, vector addition follows the triangle inequality principle, which states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides. Similarly, in vector addition, the magnitude of the resultant vector must be equal to or greater than the sum of the magnitudes of the original vectors.
Mathematically, if vector A has magnitude A and vector B has magnitude B, then the magnitude of the resultant vector R can be found using the following equation:
|R| ≤ |A| + |B|
This inequality ensures that the magnitude of the resultant vector is always equal to or greater than the magnitudes of the original vectors. Therefore, the magnitude of the resultant vector cannot be smaller than the magnitude of either of the vectors.
Yes, the magnitude of the resultant of two vectors can be smaller than the magnitude of either of them. This happens when the two vectors are in opposite directions and their magnitudes are large enough such that the magnitude of their resultant becomes smaller.
To understand this concept, let's consider two vectors A and B. The magnitude of vector A is denoted by |A| and the magnitude of vector B is denoted by |B|. In order to find the resultant vector, we can use the vector addition principle.
When the two vectors are in the same direction, their magnitudes add up and the resultant vector has a magnitude greater than either of them. Mathematically, it can be represented as |A + B| = |A| + |B|.
However, when the two vectors are in opposite directions, the magnitudes subtract from each other. In this case, the magnitude of the resultant vector becomes smaller than either of the individual vectors. Mathematically, it can be represented as |A - B| = |A| - |B|, assuming |A| > |B|.
For example, let's say vector A has a magnitude of 5 units and vector B has a magnitude of 3 units. If they are in opposite directions, their resultant vector can be found by subtracting the smaller magnitude from the larger magnitude: |A - B| = |5 - 3| = 2 units. Here, the magnitude of the resultant vector (2 units) is smaller than the magnitude of either vector A (5 units) or vector B (3 units).
So, in summary, the magnitude of the resultant vector can indeed be smaller than the magnitude of either of the individual vectors when they are in opposite directions.