Class 12 Vector Algebra · Long Answer3 Marksvector_identities
Class 12 Maths · Vector Algebra · Long Answer #64
Nonzero vectors satisfy a×b=c×d and a×c=4b×d, with a≠2d and c≠2b. Prove a−2d is parallel to 2b−c.
✅ Step-by-Step Solution
Final Answer: (a−2d) ∥ (2b−c)
- Expand (a−2d)×(2b−c).
- This equals 2a×b−a×c−4d×b+2d×c.
- Use the hypotheses and antisymmetry: 2c×d−4b×d+4b×d−2c×d=0.
- The two factors are nonzero by assumption; zero cross product proves they are parallel.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Expand (a−2d)×(2b−c).
- [1 mark] This equals 2a×b−a×c−4d×b+2d×c.
- [1 mark] Use the hypotheses and antisymmetry: 2c×d−4b×d+4b×d−2c×d=0.
- [0.5 marks] The two factors are nonzero by assumption; zero cross product proves they are parallel.
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