Question 2.122: Determine the angle θ between cables AB and AC .
Determine the angle θ between cables AB and AC .
Position Vectors: The position vectors and must be determined first.
From Fig. a
r _{A C}=\left(5 \cos 60^{\circ}-0\right) i +(0-6) j +\left(5 \sin 60^{\circ}-2\right) k =\{2.5 i -6 j +2.330 k \} ft
The magnitudes of r _{A B} and r _{A C} are
r _{A B}=\sqrt{(-3)^2+(-6)^2}=7 ftr _{A C}=\sqrt{2.5^2+(-6)^2+2.330^2}=6.905 ft
Vector Dot Product:
\begin{aligned}r _{A B} \cdot r _{A C} &=(-3 i -6 j +2 k ) \cdot(2.5 i -6 j +2.330 k ) \\&=(-3)(2.5)+(-6)(-6)+(2)(2.330) \\&=33.160 ft ^2\end{aligned}Thus
\theta=\cos ^{-1}\left(\frac{ r _{A B} \cdot r _{A C}}{ r _{A B} r _{A C}}\right)=\cos ^{-1}\left[\frac{33.160}{7(6.905)}\right]=46.7^{\circ}
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