Class 12 Vector Algebra · Short Answer2 Marksorthogonality
Class 12 Maths · Vector Algebra · Short Answer #46
Prove that for nonzero a,b, |a + b| = |a − b| if and only if a ⟂ b.
✅ Step-by-Step Solution
Final Answer: |a + b| = |a − b| ⇔ a · b = 0 ⇔ a ⟂ b.
- Expand |a + b|² = |a|² + |b|² + 2a · b and |a − b|² = |a|² + |b|² − 2a · b.
- Their difference is 4a · b. Thus equality of magnitudes implies a · b = 0. Conversely, if a · b = 0, the squared magnitudes coincide; nonnegative square roots give equality.
- Because a,b are nonzero, a · b = 0 is equivalent to perpendicularity. Both implications are proved.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Expand |a + b|² = |a|² + |b|² + 2a · b and |a − b|² = |a|² + |b|² − 2a · b.
- [1 mark] Their difference is 4a · b. Thus equality of magnitudes implies a · b = 0. Conversely, if a · b = 0, the squared magnitudes coincide; nonnegative square roots give equality.
- [0.5 marks] Because a,b are nonzero, a · b = 0 is equivalent to perpendicularity. Both implications are proved.
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