Class 12 Vector Algebra · Long Answer3 Marksvector_identities
Class 12 Maths · Vector Algebra · Long Answer #120
Prove |a−b|≤|a|+|b| for all real Euclidean vectors a,b.
✅ Step-by-Step Solution
Final Answer: |a−b|≤|a|+|b|
- Expand |a−b|²=|a|²+|b|²−2a·b.
- Cauchy–Schwarz implies a·b≥−|a||b|.
- Therefore |a−b|²≤|a|²+|b|²+2|a||b|=(|a|+|b|)².
- Both sides are nonnegative; taking square roots proves the inequality.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Expand |a−b|²=|a|²+|b|²−2a·b.
- [1 mark] Cauchy–Schwarz implies a·b≥−|a||b|.
- [1 mark] Therefore |a−b|²≤|a|²+|b|²+2|a||b|=(|a|+|b|)².
- [0.5 marks] Both sides are nonnegative; taking square roots proves the inequality.
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