Final answer:
The magnitude of vector c (the sum of vectors a and b) is less than or equal to the sum of the magnitudes of a and b, and always greater than or equal to the difference between their magnitudes, depending on the angle between the two vectors.
Step-by-step explanation:
If a vector c is defined to be the sum of two vectors a and b (i.e., c = a + b), the following statements can be made about the magnitude of vector c:
- Statement 1) The magnitude of c is not necessarily equal to the sum of the magnitudes of a and b.
- Statement 2) The magnitude of c is not greater than the sum of the magnitudes of a and b.
- Statement 3) The magnitude of c is less than or equal to the sum of the magnitudes of a and b (Triangle Inequality).
- Statement 4) The magnitude of c is not equal to the difference between the magnitudes of a and b.
- Statement 5) The magnitude of c is always greater than or equal to the difference between the magnitudes of a and b (Reverse Triangle Inequality).
- Statement 6) The magnitude of c is not necessarily less than the difference between the magnitudes of a and b; it depends on the vectors' directions.
Therefore, statements 3) and 5) are true. When two vectors are added, the resulting vector's magnitude depends on both the magnitudes of the individual vectors and the angle between them. If the vectors are in the same direction, their magnitudes add up directly; if they are in opposite directions, the magnitude of the resultant is the difference in magnitudes; if they are at some other angle, the resultant magnitude is less than the sum of the individual magnitudes.