Which of the following statements on proximity measures is FALSE?AIf the Euclidean distance of 2 vectors is zero, the cosine similarity is also always zeroBIf the cosine similarity of 2 vectors is zero, the Euclidean distance is also always zeroCIf the Manhattan distance of 2 vectors is zero, the cosine similarity is also always zeroDAll of the choices
Question
Which of the following statements on proximity measures is FALSE?AIf the Euclidean distance of 2 vectors is zero, the cosine similarity is also always zeroBIf the cosine similarity of 2 vectors is zero, the Euclidean distance is also always zeroCIf the Manhattan distance of 2 vectors is zero, the cosine similarity is also always zeroDAll of the choices
Solution
The false statement among the options is B: "If the cosine similarity of 2 vectors is zero, the Euclidean distance is also always zero".
Here's why:
A) If the Euclidean distance of 2 vectors is zero, it means the two vectors are identical. Therefore, the cosine similarity is 1, not zero. This statement is true.
B) If the cosine similarity of 2 vectors is zero, it means the vectors are orthogonal or not related. However, this does not imply that the Euclidean distance between them is zero. The Euclidean distance could be any value depending on the magnitude of the vectors. So, this statement is false.
C) If the Manhattan distance of 2 vectors is zero, it means the two vectors are identical. Therefore, the cosine similarity is 1, not zero. This statement is true.
D) Since statement B is false, the option "All of the choices" is also false.
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