A particle moves such that its position vector r → t = cos ⁡ ω t i ^ + sin ⁡ ω t…

A particle moves such that its position vector r→t=cos⁡ωti^+sin⁡ωtj^ where ω is a constant and t is time. Then which of the following statements is true for the velocity v→t and acceleration a→t of the particle:
  1. v→ is perpendicular to r→ and a→ is directed away from the origin
  2. v→ and a→ both are perpendicular to r→
  3. v→ and a→ both are parallel to r→
  4. v→ is perpendicular to r→ and a→ is directed towards the origin

Solution

r→=cos⁡ωti^+sin⁡ωtj^
v→=dr→dt=ω-sin⁡ωti^+cos⁡ωtj^
a→=dv→dt=-ω2cos⁡ωti^+sin⁡ωtj^
a→=-ω2r→∴a→ is antiparallel to r→
v→.r→=ω-sin⁡ωtcos⁡ωt+cos⁡ωtsin⁡ωt=0
So v→⊥r→

Asked in: JEE Main 2020 (08 Jan Shift 2)

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