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Let [(A)\vec] = (2, 37 °) and [(B)\vec] = (6, 269 °), where the angles are measured counterclockwise from the positive x-axis. What is [(C)\vec] = 4 [(A)\vec] + [(B)\vec]? Give your answer in component form.

Question

Let [(A)\vec] = (2, 37 °) and [(B)\vec] = (6, 269 °), where the angles are measured counterclockwise from the positive x-axis. What is [(C)\vec] = 4 [(A)\vec] + [(B)\vec]? Give your answer in component form.

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Solution

Sure, let's solve this step by step.

Step 1: Convert the polar coordinates to Cartesian coordinates.

For vector A: x = rcos(θ) = 2cos(37°) = 1.6 y = rsin(θ) = 2sin(37°) = 1.2 So, A = (1.6, 1.2)

For vector B: x = rcos(θ) = 6cos(269°) = -0.6 y = rsin(θ) = 6sin(269°) = -5.8 So, B = (-0.6, -5.8)

Step 2: Multiply vector A by 4. 4A = 4*(1.6, 1.2) = (6.4

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