Class 12 Vector Algebra · Long Answer3 Marksvector_identities
Class 12 Maths · Vector Algebra · Long Answer #82
If a×b=c×d and a×c=b×d, with a≠d and b≠c, prove a−d is parallel to b−c.
✅ Step-by-Step Solution
Final Answer: (a−d) ∥ (b−c)
- Expand (a−d)×(b−c)=a×b−a×c−d×b+d×c.
- By antisymmetry, this is a×b−a×c+b×d−c×d.
- Substitute a×b=c×d and a×c=b×d; all terms cancel.
- The cross product is zero and both factors are nonzero, so they are parallel.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Expand (a−d)×(b−c)=a×b−a×c−d×b+d×c.
- [1 mark] By antisymmetry, this is a×b−a×c+b×d−c×d.
- [1 mark] Substitute a×b=c×d and a×c=b×d; all terms cancel.
- [0.5 marks] The cross product is zero and both factors are nonzero, so they are parallel.
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