Final answer:
The angle between the force and displacement vectors is approximately 139.1°.
Step-by-step explanation:
To find the angle between the force and displacement vectors, we can use the dot product formula:
F · d = |F| |d| cos(theta)
Given that the force vector F = -11.6î + 1.19 N and the displacement vector d = 3.1î - 0.49, we can plug in the values:
-11.6 * 3.1 + 1.19 * -0.49 = |F| |d| cos(theta)
Simplifying, we get:
-35.96 + (-0.5831) = |F| |d| cos(theta)
-36.5431 = |F| |d| cos(theta)
Since we know the magnitudes of both vectors (|F| = sqrt((-11.6)^2 + 1.19^2) and |d| = sqrt(3.1^2 + (-0.49)^2)), we can solve for cos(theta):
cos(theta) = -36.5431 / (|F| |d|)
Plugging in the values:
cos(theta) = -36.5431 / (sqrt((-11.6)^2 + 1.19^2) * sqrt(3.1^2 + (-0.49)^2))
cos(theta) = -36.5431 / (14.97329 * 3.147063)
cos(theta) = -36.5431 / 47.064157
cos(theta) = -0.77607
Now we can solve for theta by taking the inverse cosine of -0.77607:
theta = acos(-0.77607)
theta ≈ 139.1°