If the magnitude of the sum of two vectors is greater than the magnitude of either vector, then:
If the magnitude of the sum of two vectors is less than the magnitude of either vector, then:
a. the scaler product of the vectors must be negative
b. the scaler product of the vectors must be positive
c. the vectors must be parallel and in opposite directions
d. the vectors must be parallel and in the same direction
e. none of the above.
Are you supposed to pick an answer to go with each of the first two statements? The magnitude of the vector sum of A and B must be in the range between |A| + |B| and |A| - |B| (assuming |A| is larger than of the same as |B|)
(a) through (d) are not true in either case.
To determine the correct answer for each statement, let's analyze the conditions given:
Statement 1: "If the magnitude of the sum of two vectors is greater than the magnitude of either vector."
In this case, we can conclude that the vectors must not be parallel or in the same direction, as adding two vectors in the same direction will result in a sum with a magnitude greater than either vector. Therefore, we can eliminate options (c) and (d).
Next, we need to consider the scalar product (also known as dot product) of the vectors. The dot product of two vectors is positive if the vectors are either parallel or have an acute angle between them. Since the vectors cannot be parallel according to our previous conclusion, the only possibility is that the vectors have an acute angle.
Hence, the correct answer for Statement 1 is (b) "the scalar product of the vectors must be positive."
Statement 2: "If the magnitude of the sum of two vectors is less than the magnitude of either vector."
In this case, since the magnitude of the sum is less than either vector, we can conclude that the vectors must be either parallel and in opposite directions, or in the case of non-parallel vectors, they must have an obtuse angle between them.
Hence, the correct answer for Statement 2 is (e) "none of the above," as none of the given options match the conditions.
Therefore, the answers to the statements are:
Statement 1: (b) "the scalar product of the vectors must be positive."
Statement 2: (e) "none of the above."
Yes, you are correct. None of the options (a) through (d) are true in either case. The correct answer is (e) none of the above.
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