Class 12 Vector Algebra · Long Answer3 Markscross_product_area
Class 12 Maths · Vector Algebra · Long Answer #81
Prove a parallelogram with diagonal vectors a,b has area |a×b|/2. Evaluate it for a=(2,−1,1), b=(1,3,−1).
✅ Step-by-Step Solution
Final Answer: √62/2 square units
- Let adjacent sides be p,q and orient diagonals a=p+q, b=q−p. Then p=(a−b)/2 and q=(a+b)/2.
- Compute p×q=¼(a−b)×(a+b)=½(a×b). Thus area=|p×q|=|a×b|/2.
- For the given diagonals, a×b=(−2,3,7), of squared norm 4+9+49=62.
- Therefore the area is √62/2 square units.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let adjacent sides be p,q and orient diagonals a=p+q, b=q−p. Then p=(a−b)/2 and q=(a+b)/2.
- [1 mark] Compute p×q=¼(a−b)×(a+b)=½(a×b). Thus area=|p×q|=|a×b|/2.
- [1 mark] For the given diagonals, a×b=(−2,3,7), of squared norm 4+9+49=62.
- [0.5 marks] Therefore the area is √62/2 square units.
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