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Vector Algebra

Question
CBSEENMA12034005
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Does there exist scalars u, v, w such that
        straight u space stack straight e subscript 1 with rightwards arrow on top space plus space straight v space stack straight e subscript 2 with rightwards arrow on top space plus space straight w space stack straight e subscript 3 with rightwards arrow on top space equals space straight i with hat on top

where   stack straight e subscript 1 with rightwards arrow on top space equals space straight k with hat on top comma space space space stack straight e subscript 2 with rightwards arrow on top space equals space straight j with hat on top space plus space straight k with hat on top comma space space stack straight e subscript 3 with rightwards arrow on top space equals space minus straight j with hat on top space plus space 2 space straight k with hat on top ?

Solution

Here,   stack straight e subscript 1 with rightwards arrow on top space equals straight k with hat on top comma space space stack straight e subscript 2 with rightwards arrow on top space equals space straight j with hat on top space plus space straight k with hat on top comma space space space stack straight e subscript 3 with rightwards arrow on top space equals space minus straight j with hat on top space plus space 2 space straight k with hat on top
Now comma space space straight u space stack straight e subscript 1 with rightwards arrow on top space plus space straight v space stack straight e subscript 2 with rightwards arrow on top space plus space straight w space stack straight e subscript 3 with rightwards arrow on top space equals space straight i with hat on top
rightwards double arrow space space space straight u left parenthesis straight k with hat on top right parenthesis space plus space straight v left parenthesis straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space straight w space open parentheses negative straight j with hat on top space plus space 2 space straight k with hat on top close parentheses space equals straight i with hat on top rightwards double arrow space space 0 straight i with hat on top space plus space left parenthesis straight v minus straight w right parenthesis space straight j with hat on top space plus space left parenthesis straight u plus straight v space plus space 2 straight w right parenthesis space straight k with hat on top space equals space 1 space straight i with hat on top space plus space 0 space straight j with hat on top space plus space 0 space straight k with hat on top rightwards double arrow space space 0 space equals space 1 comma space space space which space is space not space possible therefore space space space there space exist space on space scalars space straight u comma space straight v comma space straight w space such space that space straight u space stack straight e subscript 1 with rightwards arrow on top space plus space straight v space stack straight e subscript 2 with rightwards arrow on top space plus space straight w space stack straight e subscript 3 with rightwards arrow on top space equals space straight i with hat on top

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