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Question
For any vector `vec"a"`, prove that `hat"i" xx (vec"a" xx hat"i") + hat"j" xx (vec"a" xx hat"j") + hat"k" xx (vec"a" xx hat"k") = 2vec"a"`
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Solution
Let `vec"a" = "a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k"`
`vec"a" xx hat"i" = |(hat"i", hat"j", hat"k"),("a"_1, "a"_2, "a"_3),(1, 0, 0)|`
= `hat"i"(0) - hat"j"(- "a"_3) + hat"k"(0 - "a"_2)`
`hat"i" xx (vec"a" xx hat"i") = |(hat"i", hat"j", hat"k"),(1, 0, 0),(0, "a"_3, - "a"_2)|`
= `hat"i"(0) - hat"j"(- "a"_2) + hat"k"("a"_3)`
= `"a"_2hat"j" + "a"_3hat"k"`
Similarly `hat"j" xx (vec"a" xx hat"j") = "a"_1hat"i" + "a"_3hat"k"`
`hat"k" xx (vec"a" xx hat"k") - "a"_1hat"i" + "a"_2hat"j"`
`hat"i" xx (vec"a" xx hat"i") + hat"j" xx (vec"a" xx hat"j") + hat"k" xx (vec"a" xx hat"k")`
= `2"a"_1hat"i" + 2"a"_2hat"j" + 2"a"_3hat"k"`
= `2("a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k")`
= `2hat"a"`
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