Final answer:
To determine the coordinate direction angle α, use the orientation of the force vector relative to a reference frame and adjust the angle according to the quadrant it lies in. For the second and third quadrants or when measuring in the fourth quadrant, 180° is added to the angle α.
Step-by-step explanation:
To determine the coordinate direction angle α of a force vector F, we must consider the vector's orientation with respect to a reference frame. In mechanics, the coordinate direction angles, often denoted as α, β, and γ (or sometimes θ), are used to describe the angle between the force vector and the x, y, and z axes, respectively. Generally, if the vector lies in the first or fourth quadrant, where the x-component (Ax) is positive, the angle α in Equation 2.16 is identical to the direction angle A.
However, for vectors lying in the second or third quadrant where Ax is negative, the direction angle α is adjusted by adding 180°. Moreover, when measuring angles in the fourth quadrant, we may need to add 180° because the angle is measured clockwise from the positive x-axis. In practical scenarios, we may use trigonometric relationships and the Pythagorean theorem to calculate this angle rather than measuring it with a protractor.
In the given context, the provided information states the direction angles are 0 = −α = −55°, 0β = 90° − β = 30°, and 0γ = 90° + y = 145°. Applying these to the force components and the relevant equations would yield the scalar components and the direction angles of interest. It is important to note that while calculating the angle α for a vector in the fourth quadrant, 180° must be added since the initial angle is negative which indicates a clockwise measurement from the positive x-axis.