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Question
For any vector `vec"a"` prove that `|vec"a" xx hat"i"|^2 + |vec"a" xx hat"j"|^2 + |vec"a" xx hat"k"|^2 = 2|vec"a"|^2`
Solution
Let `vec"a" = "a"_1 hat"i" + "a"_2 hat"j" + "a"_3 hat"k"`
`vec"a" xx hat"i" = ("a"_1 hat"i" + "a"_2 hat"j" + "a"_3 hat"k") xx hat"i"`
= `"a"_1 hat"i" xx hat"i" + "a"_2hat"j" xx hat"i" + "a"_3hat"k" xx hat"i"`
= `"a"_1 xx 0 - "a"_2hat"k" + "a"_3hat"j"`
`vec"a" xx hat"i" = "a"_3hat"j" - "a"_2 hat"k"`
`|vec"a" xx hat"i"| = sqrt("a"_3^2 + (- "a"_2)^2`
`|vec"a" xx hat"i"|^2 = "a"_3^2 + "a"_2^2` .......(1)
`vec"a" xx hat"j" = ("a"_1 hat"i" + "a"_2 hat"j" + "a"_3 hat"k") xx hat"j"`
= `"a"_1 hat"i" xx hat"j" + "a"_2 hat"j" xx hat"j" + "a"_3 hat"k" xx hat"j"`
= `"a"_1 hat"k" + "a"_2 xx 0 - "a"_3 hat"i"`
`vec"a" xx hat"j" = "a"_1 hat"k" - "a"_3 hat"i"`
`|vec"a" xx hat"j"| = sqrt("a"_1^2 + (- "a"_3)^2`
`|vec"a" xx hat"j"|^2 = "a"_1^2 + "a"_3^2` ........(2)
`vec"a" xx vec"k" = ("a"_1 hat"i" + "a"_2 hat"j" + "a"_3 hat"k") xx hat"k"`
= `"a"_1 hat"i" xx hat"k" + "a"_2 hat"j" xx hat"k" + "a"_3 hat"k" xx hat"k"`
= `"a"_1 (- hat"j") + "a"_2 hat"i" + 0`
`vec"a" xx hat"k" = "a"_2 hat"i" - "a"_1 hat"j"`
`|vec"a" xx hat"k"| = sqrt("a"_2^2 + (- "a"_1)^2`
`|vec"a" xx hat"k"|^2 = "a"_2^2 + "a"_1^2` .......(3)
Equation (1) + (2) + (3) ⇒
`|vec"a" xx hat"i"|^2 + |vec"a" xx hat"j"|^2 + |vec"a" xx hat"k"|^2`
= `"a"_3^2 + "a"_2^2 + "a"_1^2 + "a"_3^2 + "a"_2^2 + "a"_1^2`
= `2("a"_1^2 + "a"_2^2 + "a"_3^2)` .......(4)
`|vec"a"| = |"a"_1 hat"i" + "a"_2 hat"j" + "a"_3 hat"k"|`
`|vec"a"| = sqrt("a"_1^2 + "a"_2^2 + "a"_3^2)`
`|vec"a"| = "a"_1^2 + "a"_2^2 + "a"_3^2` .......(5)
From equation (4) and (5)
`|vec"a" xx hat"i"|^2 + |vec"a" xx hat"j"|^2 + |vec"a" xx hat"k"|^2 = 2|vec"a"|^2`
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