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# The vector $(\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d})$is

(1)perpendicular to $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$(2) parallel tothevectors $(\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d})$(3) parallel to the line of intersection of the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$ and the plane containing $\overrightarrow{c}$ and $\overrightarrow{d}$(4) perpendicular to the line of intersection of the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$ and the plane containing $\overrightarrow{c}$ and $\overrightarrow{d}$

$(\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d}) = [\overrightarrow{a} \; \overrightarrow{b} \; \overrightarrow{d} ] \overrightarrow{c} - [ \overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] \overrightarrow{d}$ -----(1)
$(\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d}) = [\overrightarrow{c} \; \overrightarrow{d} \; \overrightarrow{a} ] \overrightarrow{b} - [ \overrightarrow{c} \overrightarrow{b} \overrightarrow{d}] \overrightarrow{a}$-----(2)
Equation (1) => vector $(\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d})$
lies on the plane containing $\overrightarrow{c}$ and $\overrightarrow{d}$
Equation (1) => vector $(\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d})$
lies on the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$
$\therefore (\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d})$ is a vector parallel to the lie of intersection of the plane containing the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ and the plane containing the vectors $\overrightarrow{a}$ a$\therefore (\overrightarrow{a}\times\overrightarrow{b})\times(\overrightarrow{c}\times\overrightarrow{d})$ is a vector parallel to the lie of intersection of the plane containing the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ and the plane containing the vectors $\overrightarrow{a}$ ad $\overrightarrow{b}$ and the plane containing the vectors $\overrightarrow{c}$ and $\overrightarrow{d}$d $\overrightarrow{b}$ and the plane containing the vectors $\overrightarrow{c}$ and $\overrightarrow{d}$