Question 32: The cosine similarity between Y and W is:

Story: The Shadow War – Cracking the Enigma
In the heart of World War II, deep inside Bletchley Park, a team of elite cryptanalysts—Captain Edward Sinclair, Dr. Margaret Holloway, Lieutenant James Carter, and Specialist Anna Whitmore—worked tirelessly to decode intercepted German transmissions encrypted by the Enigma machine.
One evening, they received four coded messages, each suspected to contain crucial intelligence about an upcoming U-boat assault. However, intelligence reports suggested that one of the messages was deliberately altered to mislead the Allies. The team had to determine which three messages were most aligned and which one was a decoy planted by the enemy.
Based on the above data, answer the given subquestions. Using early computing techniques and mathematical models, they converted each message into numerical vectors:
● Message X: [7, 12, 9, 5, 8]
● Message Y: [6, 11, 8, 4, 9]
● Message Z: [5, 10, 7, 6, 7]
● Message W: [8, 13, 10, 3, 9]
With time running out before the Royal Navy launched its counter-offensive, the team must quickly calculate the cosine similarity between message pairs to determine which three should be used to uncover the real enemy plans. One mistake could cost thousands of lives at sea.

The cosine similarity between Y and W is: