Class 12 Vector Algebra · Long Answer3 Markscross_product
Class 12 Maths · Vector Algebra · Long Answer #72
For a=(1,1,1), b=(2,4,−5), c=(λ,2,3), the magnitude of a crossed with the unit vector along b+c is √2. Find λ.
✅ Step-by-Step Solution
Final Answer: λ=1
- Put s=b+c=(λ+2,6,−2), with |s|²=λ²+4λ+44.
- Compute a×s=(−8,λ+4,4−λ), whose squared norm is 96+2λ².
- The condition gives (96+2λ²)/(λ²+4λ+44)=2, hence 8λ=8. The denominator is always positive.
- Therefore λ=1, which gives cross-product magnitude √2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put s=b+c=(λ+2,6,−2), with |s|²=λ²+4λ+44.
- [1 mark] Compute a×s=(−8,λ+4,4−λ), whose squared norm is 96+2λ².
- [1 mark] The condition gives (96+2λ²)/(λ²+4λ+44)=2, hence 8λ=8. The denominator is always positive.
- [0.5 marks] Therefore λ=1, which gives cross-product magnitude √2.
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