Consider two vectors where F1 = 40 N, F2 = 57 N, and \theta _{1} = 240o and \theta _{2} = 25o, measured from the positive x-axis with counter-clockwise being positive. What is the magnitude of the equilibriant?

For the equilibrant, add the same vectors but with opposite directions (180 deg away), namely

theta_{1} = 60 deg and theta_{2} = 205 deg

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To find the magnitude of the equilibriant, we need to find the resultant vector by adding the two given vectors and then find the magnitude of the resultant vector.

Let's start by calculating the components of each vector:

F1 = 40 N, θ1 = 240°
Fx1 = F1 * cos(θ1)
Fy1 = F1 * sin(θ1)

F2 = 57 N, θ2 = 25°
Fx2 = F2 * cos(θ2)
Fy2 = F2 * sin(θ2)

Now, let's calculate the x and y components separately:

Fx = Fx1 + Fx2
Fy = Fy1 + Fy2

Next, we can calculate the magnitude of the resultant vector using the Pythagorean theorem:

Magnitude of the resultant vector = sqrt(Fx^2 + Fy^2)

Substituting the values we calculated:

Magnitude of the resultant vector = sqrt((Fx1 + Fx2)^2 + (Fy1 + Fy2)^2)

Finally, we can plug in the values and calculate the magnitude of the equilibriant.