Questions: Vector A has magnitude a = 800 m and direction θ = 30.0°, whereas B has magnitude B = 400 m and θ = 210.0°. What is R = A + B? R = [ ]ˆi + [ ]ˆj m

Transcript text: Vector A has magnitude a = 800 m and direction θ = 30.0°, whereas B has magnitude B = 400 m and θ = 210.0°. What is R = A + B? R = [ ]ˆi + [ ]ˆj m

Solution

Solution Steps

Step 1: Convert Vector A to Component Form

First, we need to convert vector \( \mathbf{A} \) from its magnitude and direction to its component form. The components of vector \( \mathbf{A} \) can be found using the following formulas: \[ A_x = a \cos \theta \] \[ A_y = a \sin \theta \] Given \( a = 800 \) m and \( \theta = 30.0^\circ \): \[ A_x = 800 \cos 30.0^\circ = 800 \times 0.8660 = 692.8 \, \text{m} \] \[ A_y = 800 \sin 30.0^\circ = 800 \times 0.5000 = 400.0 \, \text{m} \]

Step 2: Convert Vector B to Component Form

Next, we convert vector \( \mathbf{B} \) to its component form using the same approach: \[ B_x = B \cos \theta \] \[ B_y = B \sin \theta \] Given \( B = 400 \) m and \( \theta = 210.0^\circ \): \[ B_x = 400 \cos 210.0^\circ = 400 \times (-0.8660) = -346.4 \, \text{m} \] \[ B_y = 400 \sin 210.0^\circ = 400 \times (-0.5000) = -200.0 \, \text{m} \]

Step 3: Add the Components of Vectors A and B

Now, we add the corresponding components of vectors \( \mathbf{A} \) and \( \mathbf{B} \) to find the resultant vector \( \mathbf{R} \): \[ R_x = A_x + B_x = 692.8 + (-346.4) = 346.4 \, \text{m} \] \[ R_y = A_y + B_y = 400.0 + (-200.0) = 200.0 \, \text{m} \]