To use the dot product to find the components of forces acting in arbitrary dire

Question

To use the dot product to find the components of forces acting in arbitrary directions. A three-legged structure is shown below. It is attached to the ground at points B = (4.46, 0, 0) and C = (-4.46, 0, 0) and connected to the ceiling at point A = (-5.70, 3-58, 5.82). The three legs connect at point D = (0, 3.06, 2.50), where two forces, F and P, act. Force F is given by F = 5.60 i + 17.5 j + 24.4 kN; P has magnitude 35.0 N and direction angles alpha = 128.5 degree, beta = 66.0 degree, and gamma = 48.0 degree for the x, y, and z axes, respectively. Determining the force along a member Determine the magnitude, F_DA, acting along member DA, due to the applied force F = 5.60 i - 17.5 j + 24.4 kN. Express your answer to three significant figures and include the appropriate units. Finding the component of a force perpendicular to a direction Given F = 5.60 i - 17.5 j + 24.4 kN, find the component of F that acts perpendicular to member DA such that the vector addition of the perpendicular and parallel components of F (F = F_ + F_) with respect to DA equals F. Express your answer in component form. Express your answers, separated by commas, to three significant figures.

Explanation / Answer

Part C

We have to find component of force F along line DA.

Now, DA = (-5.70 - 0)i + (3.58 - 3.06)j + (5.82 - 2.5)k

or, DA = -5.7i + 0.52j + 3.32k

lDAl = 6.62

So component of F along DA is,

Fll = F.(DA/lDAl) = (5.6i - 17.5j + 24.4k).(-5.7i + 0.52j + 3.32k)/6.62

or, Fll = 6.04 N is the magnitude of the force F along DA.

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Part D

Component of F perpendicular to DA is given by

F = Fll + Fperp (perp - perpendicular)

or, 5.6i - 17.5j + 24.4k = 4.82i + 1.375j - 12.242k + Fperp

or, Fperp = [0.78i - 18.875j + 36.642 k]N is the perpendicular component.

So , the answer would be, 0.78iN, - 18.875jN, 36.642 kN

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This concludes the answers. Check the answer and let me know if it's correct. If you need any more clarification, modification or correction, feel free to ask.....

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