Vector C has a magnitude of 24.6 m and points in the − y ‑ direction. Vectors A and B both have positive y ‑ components, and make angles of α = 44.9 ° and β = 27.7 ° with the positive and negative x - axis, respectively. If the vector sum A + B + C = 0 , what are the magnitudes of A and B ?


Answer:

A= 61.35

B= -44.40

Explanation:

1. Using the components method we have:

A_{x}= A cos \alpha\\B_{x}= B cos \beta\\C_{x}= 0\\\\A_{y}= A sin \alpha\\B_{y}= B sin \beta\\C_{y}= 24.6\\

Considering that the vector sum A+B+C=0, then:

|V|=\sqrt{V_{x}^{2} +V_{y}^{2} }=0

Then:

V_{x} ^{2} =0; V_{x} =0\\V_{y} ^{2} =0; V_{y} =0

It means the value of x and y component is 0.

2. Determinate the equations that describe each component:

V_{x}= A cos \alpha -B cos \beta=0  (1)\\V_{y}= A sin \alpha +B sin \beta - C=0   (2)

Form Eq. (1):

A=B \frac{cos \beta}{cos \alpha}     (3)

Replacing A in Eq. (2):

(B\frac{cos \beta}{cos \alpha})(sin \alpha)+ B sin\beta-C=0\\(B\frac{cos \beta}{cos \alpha})(sin \alpha)+ B sin\beta-=C\\\\B(\frac{cos \beta. sin \alpha}{cos \alpha}+ sin\beta)=C\\B=C(\frac{cos \beta. sin \alpha}{cos \alpha}+ sin\beta)^{-1}     (4)

Replacing values of C, α and β in (4):

B= 24.6 (\frac{(cos 27.7)(sin 44.9)}{cos 44.9}+sin 27.7)^{-1}  \\B= -44.4

Replacing value of B in (3)

A=-44.40\frac{cos 27.7}{sin 49.9} \\A= 61.35


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