A) 7
B) 16
C) \[-1\]
D) 4
E) \[-17\]
Correct Answer: D
Solution :
Given that \[(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=5\overrightarrow{c}+6\overrightarrow{d}\] \[\Rightarrow \]\[[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}]\overrightarrow{c}-[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]\overrightarrow{d}=5\overrightarrow{c}+6\overrightarrow{d}\] \[\Rightarrow \]\[[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}]=5\]and\[[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]=-6\] Now, \[\overrightarrow{a}-\overrightarrow{b}\times (\overrightarrow{a}+\text{ }\overrightarrow{c}+2\overrightarrow{d})\] \[=\overrightarrow{a}.\overrightarrow{b}\times \overrightarrow{c}+2(\overrightarrow{a}\text{.}\overrightarrow{b}\times \overrightarrow{d})\] \[=[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]+2[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}]\] \[=-6+2\times 5=10-6=4\]
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