If $straight a with rightwards arrow on top$ is any vector, then show that
$straight a with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top space minus space straight i with hat on top right parenthesis space straight i with hat on top space plus space left parenthesis straight a with rightwards arrow on top. space straight j with hat on top right parenthesis space straight j with hat on top space plus space left parenthesis straight a with rightwards arrow on top space. space straight k with hat on top right parenthesis space straight k with hat on top.$

Solution

straight a with hat on top space equals space straight a subscript 1 space straight i with hat on top space plus space straight a subscript 2 space straight j with hat on top space plus space straight a subscript 3 space straight k with hat on top

...(1)

straight a with hat on top. space straight i with hat on top space equals straight a subscript 1 space straight i with hat on top. space straight i with hat on top space plus space straight a subscript 2 space straight j with hat on top space. space straight i with hat on top space plus space straight a subscript 3 space straight k with hat on top space. space straight i with hat on top space equals space straight a subscript 1 left parenthesis 1 right parenthesis space plus space straight a subscript 2 left parenthesis 0 right parenthesis space plus space straight a subscript 3 left parenthesis 0 right parenthesis straight d with rightwards arrow on top space space straight i with hat on top space equals space straight a subscript 1

Similarly,

straight a with rightwards arrow on top. space straight j with hat on top space equals straight a subscript 2 comma space straight a with rightwards arrow on top space. straight k with hat on top space equals space straight a subscript 3

from (1), we get

straight a with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top. space straight i with hat on top right parenthesis space straight i with hat on top space plus space left parenthesis straight a with rightwards arrow on top. space straight j with hat on top right parenthesis space straight j with hat on top space plus space left parenthesis straight a with rightwards arrow on top. space straight k with hat on top right parenthesis space straight k with hat on top

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