(Vector algebra with initial and final positions) At time

PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials

Problem

(Vector algebra with initial and final positions) At time t1t_1, a particle is at position r1=4 ı^−3 ȷ^ m\mathbf r_1 = 4\,\hat\imath - 3\,\hat\jmath\,\text{m}. At a later instant t2t_2, it is at r2=−2 ı^+5 ȷ^ m\mathbf r_2 = -2\,\hat\imath + 5\,\hat\jmath\,\text{m}. (a) Determine the displacement vector Δr\Delta\mathbf r for the interval t1t_1 to t2t_2. (b) Prove that the magnitude of the displacement found in part (a) can be expressed in terms of the coordinate differences as ∣Δr∣=(x2−x1)2+(y2−y1)2|\Delta\mathbf r| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. (c) Show that Δr⋅(r2+r1)=∣r2∣2−∣r1∣2\Delta\mathbf r \cdot (\mathbf r_2 + \mathbf r_1) = |\mathbf r_2|^{2} - |\mathbf r_1|^{2}. Interpret this equality geometrically.

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