Two vectors a and b each have magnitude 1/sqrt(2) and their sum |a + b| = 1. Find |a - b|.

Correct answer: 1

Solution

Given |a| = |b| = 1/sqrt(2) and |a+b| = 1. Step 1: |a+b|² = |a|² + |b|² + 2(a.b) = 1/2 + 1/2 + 2(a.b) = 1 + 2(a.b) = 1² = 1. So 2(a.b) = 0, meaning a.b = 0 (the vectors are perpendicular). Step 2: |a-b|² = |a|² + |b|² - 2(a.b) = 1/2 + 1/2 - 0 = 1. So |a-b| = 1.

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