Question 2.123: Determine the angle Φ between cable AC and strut AO.
Determine the angle Φ between cable AC and strut AO.
Position Vectors: The position vectors r _{A C} andr _{A O} must be determined first.
From Fig. a,
r _{A O}=(0-0) i +(0-6) j +(0-2) k =\{-6 j -2 k \} f t
The magnitudes of r _{A C} andr _{A O} are
r _{A C}=\sqrt{2.5^2+(-6)^2+2.330^2}=6.905 ftr _{A O}=\sqrt{(-6)^2+(-2)^2}=\sqrt{40 ft }
Vector Dot Product:
\begin{aligned}r _{A C} \cdot r _{A O} &=(2.5 i -6 j +2.330 k ) \cdot(-6 j -2 k ) \\&=(2.5)(0)+(-6)(-6)+(2)(2.330)(-2) \\&=31.34 ft ^2\end{aligned}Thus,
\phi=\cos ^{-1}\left(\frac{ r _{A C} \cdot r _{A O}}{ r _{A C} r _{A O}}\right)=\cos ^{-1}\left[\frac{31.34}{6.905 \sqrt{40}}\right]=44.1^{\circ}
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