Let two non-collinear unit vectors a and b ^ form an acute angle. A point P moves so that at any time t the position vector OP (where, O is the origin) is given by a cos t + b ^ sin t . When P is fart

Question

Let two non-collinear unit vectors a and b ^ form an acute angle. A point P moves so that at any time t the position vector OP (where, O is the origin) is given by a cos t + b ^ sin t . When P is farthest from origin O , let M be the length of OP and u ^ be the unit vector along OP . Then,

TestHub · Accepted Answer

OP = a ^ cos t + b ^ sin t ​ ⇒ ∣ OP ∣ = ( a ^ ⋅ a ^ cos 2 t + b ^ ⋅ b ^ sin 2 t + 2 a ^ ⋅ b ^ sin t cos t ) ​ ⇒ ∣ OP ∣ = 1 + 2 a ^ ⋅ b ^ ⋅ sin t cos t ​ ⇒ ∣ OP ∣ = 1 + a ^ ⋅ b ^ ⋅ sin 2 t ​ ⇒ ∣ OP ∣ m a x ​ = 1 + a ^ ⋅ b ^ ​ at sin 2 t = 1 ⇒ t = 4 π ​ ⇒ OP ( at t = 4 π ​ ) = 2 ​ 1 ​ ( a ^ + b ^ ) ​ ∴ Unit vector along OP at ( t = 4 π ​ ) = ∣ a ^ + b ^ ∣ a ^ + b ^ ​

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