Question

If a→,b→,c→ are non-coplanar vectors and λ is a real number, then[λ(a→+b→)λ2b→λc→]=[a→b→+c→b→] for

• A. (A) No value of λ
• B. (B) Exactly one value of λ
• C. (C) Exactly two values of λ
• D. (D) Exactly three values of λ

Answer 1

Answer: (A)

(A) No value of λ

Answer 2

Explanation:
Given:Non- coplanar vectors a→,b→,c→and [λ(a→+b→)λ2b→λc→]=[a→b→+c→b→]We have to find the value of λ.since, a→,b→,c→ are non-coplanar vectors⇒[a→b→c→]≠0Now, [λ(a→+b→)λ2b→λc→]=[a→b→+c→b→]Using the definition of scalar triple product, we getλ(a→+b→)⋅(λ2b→×λc→)=a→⋅b→+c→)×b→)=a→⋅(b→×b→+c→×b→)=λ(a→+b→)⋅(λ2b→×λc→)=a→⋅(0+c→×b→)a→⋅(c→×b→)[Using properties of cross product-2]⇒λ4(a→⋅(b→×c→))+b→⋅(b→×c→)=a→⋅(c→×b→)⇒λ4([a→b→c→]+[b→b→c→])=−[a→b→c→][Using properties of scalar triple product-3]⇒λ4([a→b→c→])=−[a→b→c→]⇒λ4=−1Which is not true for any real value of λ.Hence, the correct option is (A).