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Vector Algebra

Question
CBSEENMA12034234
Wired Faculty App

If straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top be any three vectors, then straight a with rightwards arrow on top cross times space open parentheses straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top close parentheses space space equals space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top.

Solution

Let straight a with rightwards arrow on top space equals space straight a subscript 1 straight i with hat on top space plus straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top comma space space straight b with rightwards arrow on top space equals space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top comma space straight c with rightwards arrow on top space equals straight c subscript 1 straight i with hat on top space plus space straight c subscript 2 straight j with hat on top space plus space straight c subscript 3 straight k with hat on top
L.H.S. = straight a with rightwards arrow on top space cross times space left parenthesis straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top right parenthesis
            equals space left parenthesis straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top right parenthesis space cross times space open curly brackets left parenthesis straight b subscript 1 space plus space straight c subscript 1 right parenthesis space straight i with hat on top space plus left parenthesis straight b subscript 2 plus straight c subscript 2 right parenthesis space straight j with hat on top space plus left parenthesis straight b subscript 3 plus straight c subscript 3 right parenthesis space straight k with hat on top close curly brackets equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 plus straight c subscript 1 end cell cell straight b subscript 2 plus straight c subscript 2 end cell cell straight b subscript 3 plus straight c subscript 3 end cell end table close vertical bar space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight b subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell end table close vertical bar plus space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row cell straight c subscript 1 end cell cell straight b subscript 2 end cell cell straight b subscript 3 end cell end table close vertical bar space equals straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space straight R. straight H. straight S.

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