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Question

If the vector p=(a+1)^i+a^j+a^k,q=a^i+(a+1)^j+a^k and r=a^i+a^j+(a+1)^k, (aR) are coplanar and 3(p.q)2λ|r×q|2=0, then value of` λ is

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Solution

As p,q,r are coplanar,
∣ ∣a+1aaaa+1aaaa+1∣ ∣=0

R1R1+R2+R3

∣ ∣3a+13a+13a+1aa+1aaaa+1∣ ∣=0

(3a+1)∣ ∣111aa+1aaaa+1∣ ∣=0

C2C2C1 , C3C3C1

(3a+1)∣ ∣100a10a01∣ ∣=0
3a+1=0a=13

p=13(2^i^j^k),q=13(^i+2^j^k),
r=13(^i^j+2^k)

r×q=19∣ ∣ ∣^i^j^k112121∣ ∣ ∣

r×q=19(3^i3^j3^k)
=13(^i+^j+^k)

|r×q|2=13
p.q=19(22+1)=13

3(p.q)2λ|r×q|2=013λ×13=0
λ=1

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