Question 2.134: (a) What is the cross product rOA × rOB? (b) Determine a uni......

Engineering Mechanics Statics – Solution Manual [1686042]

(a) What is the cross product $r_{OA}$ × $r_{OB}$ ? (b) Determine a unit vector e that is perpendicular to $r_{OA}$ and $r_{OB}$ .

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The two radius vectors are

r _{O B}=4 i +4 j -4 k , \quad r _{O A}=6 i -2 j +3 k

(a) The cross product is

\begin{aligned}r _{O A} \times r _{O B}=\left|\begin{array}{ccc} i & j & k \\6 & -2 & 3 \\4 & 4 & -4\end{array}\right| & = i (8-12)- j (-24-12) \\& + k (24+8) \\= & -4 i +36 j +32 k~\left(m ^2\right)\end{aligned}

The magnitude is

\left| r _{O A} \times r _{O B}\right|=\sqrt{4^2+36^2+32^2}=48.33~m ^2

(b) The unit vector is

e = \pm\left(\frac{ r _{O A} \times r _{O B}}{\left| r _{O A} \times r _{O B}\right|}\right)= \pm(-0.0828 i +0.7448 j +0.6621 k )

(Two vectors.)

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