Class 12 Vector Algebra · Long Answer3 Marksunit_vectors
Class 12 Maths · Vector Algebra · Long Answer #65
Let a=(1,1,1), b=(2,4,−5), c=(λ,2,3). If a dotted with the unit vector along b+c equals 1, find λ and that unit vector.
✅ Step-by-Step Solution
Final Answer: λ=1; unit vector=(3,6,−2)/7
- Write s=b+c=(λ+2,6,−2); |s|=√((λ+2)²+40).
- The condition is (λ+6)/√((λ+2)²+40)=1, so λ+6 must be positive.
- Square: (λ+6)²=(λ+2)²+40, giving 8λ=8 and λ=1. This satisfies the unsquared equation.
- Then s=(3,6,−2), |s|=7 and the required unit vector is (3,6,−2)/7.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Write s=b+c=(λ+2,6,−2); |s|=√((λ+2)²+40).
- [1 mark] The condition is (λ+6)/√((λ+2)²+40)=1, so λ+6 must be positive.
- [1 mark] Square: (λ+6)²=(λ+2)²+40, giving 8λ=8 and λ=1. This satisfies the unsquared equation.
- [0.5 marks] Then s=(3,6,−2), |s|=7 and the required unit vector is (3,6,−2)/7.
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