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Position vector, r = (a cos ωt)i + (a sin ωt)j. Velocity vector, v = d(r)/dt = d/dt {(a cos ωt)i + (a sin ωt)j} = (−aω sin ωt)i + (aω cos ωt)j = ω{(−sin ωt)i + (cos ωt)j}. Slope of position vector = a sin ωt/a cos ωt = tan ωt. Slope of velocity vector = −a cos ωt/a sin ωt = −1/tan ωt. ∴ velocity is perpendicular to the displacement.

1. (a) Position vector, r = (a cos ωt)i + (a sin ωt)j
2. (b) Velocity vector, v = d(r)/dt = d/dt {(a cos ωt)i + (a sin ωt)j}
3. (c) Slope of position vector = a sin ωt/a cos ωt = tan ωt
4. (d) Slope of velocity vector = −a cos ωt/a sin ωt = −1/tan ωt

Correct answer: (d) Slope of velocity vector = −a cos ωt/a sin ωt = −1/tan ωt

Solution

The slope of the velocity vector is correctly calculated as −a cos ωt / a sin ωt = −1 / tan ωt, which confirms that the velocity is perpendicular to the displacement. This is the key conclusion of the problem.

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