The cable at the end of the crane boom exerts a force of 250 lb on the boom as shown. Express F as a Cartesian vector.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
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To do this question, you must remember the following equation:
(Where \alpha,\beta,\gamma are the angles between the positive x,y, and z axes respectively.)
From the diagram, we know the values of \alpha and \beta.
Subsituting the values we know into our equation gives us:
\cos^2\gamma = 0.133
(take the square root of both sides)
\cos\gamma = \pm 0.365
\gamma = \cos^{-1}(\pm 0.365)
\gamma = 68.6^0 or 111^0
From the diagram, we can see that the angle between the z-axis and the force must be larger than 90^0 because it lies in the negative quadrant. Therefore:
We can now express the force in Cartesian vector form:
F\,=\,\left\{217i+85j-90k\right\} lb
why is -90k
it should be +90k
Cos(111°) = -0.358. Also, looking at the diagram, we can see that the force is below the z-axis, so even without looking at numbers, we can see it’ll be negative.