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Vector Algebra

Question
CBSEENMA12036280
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Let top enclose straight a space comma top enclose straight b space and space top enclose straight c space be non-zero vectors such that left parenthesis top enclose straight a space straight x space top enclose straight b right parenthesis space straight x space top enclose straight c space equals space 1 third space vertical line straight b with bar on top vertical line vertical line straight c with bar on top vertical line vertical line straight a with bar on top vertical line If θ is the acute angle between the vectors top enclose straight b space and space top enclose straight c comma then sin θ equals

  • 1/3

  • fraction numerator square root of 1 over denominator 3 end fraction

  • 2/3

  • fraction numerator 2 square root of 2 over denominator 3 end fraction

Solution

D.

fraction numerator 2 square root of 2 over denominator 3 end fraction 1 third space vertical line straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line vertical line straight a with rightwards arrow on top vertical line space equals left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x space straight c with rightwards arrow on top we space know space that space left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x space straight c with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis straight b with rightwards arrow on top space minus space left parenthesis straight b with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis straight a with rightwards arrow on top 1 third space vertical line straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line straight a with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis. straight b with rightwards arrow on top space minus space left parenthesis straight b with rightwards arrow on top space. space straight c with rightwards arrow on top right parenthesis. straight a with rightwards arrow on top On space comparing comma space we space get 1 third space vertical line straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line space equals negative space left parenthesis straight b with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis space and space space left parenthesis straight a with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis rightwards double arrow space 1 third bc space equals negative space bc space cos space straight theta cos space space equals space fraction numerator negative 1 over denominator 3 end fraction cos squared space straight theta space equals space 1 third rightwards double arrow 1 minus sin squared space straight theta space equals space 1 over 9 sin squared space straight theta space equals space 1 minus 1 over 9 space equals 8 over 9 space rightwards double arrow space sin space straight theta space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction

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