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Q. Let ˆa and ˆb be two unit vectors such that |(ˆa+ˆb)+2(ˆa׈b)|=2. If θ(0,π) is the angle between ˆa and ˆb, then among the statements :
(S1):2|ˆa׈b|=|ˆaˆb|
(S2) : The projection of ˆa on (ˆa+ˆb) is 12

JEE MainJEE Main 2022Vector Algebra

Solution:

|(ˆa+ˆb)+2(ˆa׈b)|=2,θ(0,π)
((ˆa+ˆb)+2(ˆa׈b))((ˆa+ˆb)+2(ˆa׈b))=4
|ˆa+ˆb|2+4|(ˆa׈b)|2+0=4
Let the angle be θ between ˆa and ˆb
2+2cosθ+4sin2θ=4
2+2cosθ4cos2θ=0
Let cosθ=t then
2t2t1=0
2t22t+t1=0
2t(t1)+(t1)=0
(2t+1)(t1)=0
t=12 or t=1
cosθ=12
θ=2π3
not possible as θ(0,π)
Now
S12|a×b|=2sin(2π3)
|ˆaˆb|=1+12cos(2π3)
=22×(12)
=3
S1 is correct.
S2 projection of ˆa on (ˆa+ˆb).
ˆa(ˆa+ˆb)|ˆa+ˆb|=1+cos(2π3)2+2cos2π3
=1121
=12