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24 December, 16:34

Vector A → has magnitude 8.78 m at 37.0 ∘ from the + x axis. Vector B → has magnitude 8.26 m at 135.0 ∘ from the + x axis. Vector C → has magnitude 5.65 m at 210.0 ∘ from the + x axis. Using the component method, calculate the magnitude of the resultant vector

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  1. 24 December, 19:39
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    R = ( - 3.72î + 8.29j)

    Magnitude of R = 9.09 m

    Explanation:

    Let î and j represent unit vectors along the x and y axis respectively.

    Vector A - -> magnitude 8.78 m, direction 37.0° from the + x-axis

    Let the x and y components of this vector be Aₓ and Aᵧ

    A = (Aₓî + Aᵧj) m

    The components given magnitude and direction from the + x-axis are calculated as

    Aₓ = A cos θ and Aᵧ = A sin θ

    Aₓ = (8.78 cos 37°) = 7.01 m

    Aᵧ = (8.78 sin 37°) = 5.28 m

    A = (7.01î + 5.28j) m

    Vector B has magnitude 8.26 m and direction 135° from the + x-axis

    B = (Bₓî + Bᵧj) m

    Bₓ = (8.26 cos 135°) = - 5.84 m

    Bᵧ = (8.26 sin 135°) = 5.84 m

    B = (-5.84î + 5.84j) m

    Vector C has magnitude 5.65 m and direction 210° from the + x-axis

    C = (Cₓî + Cᵧj) m

    Cₓ = (5.65 cos 210°) = - 4.89 m

    Cᵧ = (5.65 sin 210°) = - 2.83 m

    C = ( - 4.89î - 2.83j) m

    The resultant force is a vector sum of all the forces. Let the resultant force be R

    R = (Rₓî + Rᵧj) m

    R = A + B + C = (7.01î + 5.28j) + (-5.84î + 5.84j) + ( - 4.89î - 2.83j)

    Summing the î and j components seperately,

    R = ( - 3.72î + 8.29j) m

    To get its magnitude,

    Magnitude of R = √ (Rₓ² + Rᵧ²) = √ ((-3.72) ² + (8.29) ²) = 9.09 m
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