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Question
Two non-collinear unit vectors →a and →b are such that [→r →a →b]=74 and →r(4→a+5→b)=0.
Let ∫−52→r.→b−2→r.→ax+1x2+1dx=π2, angle between →a and →b is such that |→r|2−34∣∣→a×→b∣∣2 attains maximum value, can be represented as a linear combination of →a and →b as
Two non-collinear unit vectors →a and →b are such that [→r →a →b]=74 and →r(4→a+5→b)=0.
Let ∫−52→r.→b−2→r.→ax+1x2+1dx=π2, angle between →a and →b is such that |→r|2−34∣∣→a×→b∣∣2 attains maximum value, can be represented as a linear combination of →a and →b as
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Solution
The correct option is D 12→a−25→b+74(→a×→b)
For attaining maximum value,
→r=(→r.→a)→a+(→r.→b)→b+[→r→a→b](→a×→b)
4→r.→a+5→r.→b=0⇒→r.→b=−45→r.→a
∫2→r.→a−2→r.→ax+1x2+1dx=∫2→r.→a−2→r.→axdxx2+1+∫2→r.→a−2→r.→adxx2+1
2∫2→r.→a0dxx2+1=π2
⇒ tan−1(2→r.→a)=π4
⇒ 2→r.→a=1.
⇒ →r.→a=12.
→r.→b=−45.12=−25
→r.(→a×→b)=74
→r=12→a−25→b+74→a×→b
For attaining maximum value,
→r=(→r.→a)→a+(→r.→b)→b+[→r→a→b](→a×→b)
4→r.→a+5→r.→b=0⇒→r.→b=−45→r.→a
∫2→r.→a−2→r.→ax+1x2+1dx=∫2→r.→a−2→r.→axdxx2+1+∫2→r.→a−2→r.→adxx2+1
2∫2→r.→a0dxx2+1=π2
⇒ tan−1(2→r.→a)=π4
⇒ 2→r.→a=1.
⇒ →r.→a=12.
→r.→b=−45.12=−25
→r.(→a×→b)=74
→r=12→a−25→b+74→a×→b
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