Question

Vector c has a magnitude 24.6 m and is in the direction of the negative y-axis. vectors a and b are at angles α = 41.4° and β = 27.7° up from the x-axis respectively. if the vector sum a b c = 0, what are the magnitudes of a and b?

Answer 1

The vector c has a magnitude of 24.6m and it is in the negative y direction. Therefore
$\vec{c} = - 24.6 \hat{j}$

The vector b is 41.4° up from the x-axis. Therefore
$\vec{b} = b[cos(41.4^{o}) \hat{i} + sin(41.4^{o}) \hat{j} ] =b(0.75\hat{i} + 0.6613 \hat{j})$

The vector a is 27.7° up from the x-axis. Therefore
$\vec{a} = a[cos(22.7^{o})\hat{i} + sin(27.7^{o})\hat{j}] = a(0.8854\hat{i} + 0.4648\hat{j})$

Because $\vec{a} +\vec{b} + \vec{c} = 0$, the sum of the x and y components should be zero. Therefore,
For the x-component,
0.8854a + 0.75b = 0
 or
 a + 0.847b = 0                            (1)
For the y-component,
0.4648a + 0.6613b - 24.6 = 0
 or
 a + 1.4228b = 52.926                (2)

Subtract (1) from (2).
0.5758b = 52.926
b = 91.917
a = -0.847b = -77.854

Answer:
The magnitude of vector a is -77.85 m
The magnitude of vector b is 91.92 m