Given the expression (a + b) . (a - b) = 198. Using the distributive property of the dot product, we get: $$|\vec{a}|^2 - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - |\vec{b}|^2 = 198$$ Since the dot product is commutative, a . b = b . a, the middle terms cancel out: $$|\vec{a}|^2 - |\vec{b}|^2 = 198$$
We are given |a| = 10|b|. Substitute this into the equation derived in Step 1: $$(10|\vec{b}|)^2 - |\vec{b}|^2 = 198$$ $$100|\vec{b}|^2 - |\vec{b}|^2 = 198$$ $$99|\vec{b}|^2 = 198$$
Divide both sides by 99: $$|\vec{b}|^2 = \frac{198}{99}$$ $$|\vec{b}|^2 = 2$$ $$|\vec{b}| = \sqrt{2}$$
Final Answer: |b| = \sqrt{2}
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