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If the vectors `3 vec p + vec q`; `5 vec p - 3 vecq` and `2 vec p + vec q`; `4 vec p - 2 vec q` are pairs of mutually perpendicular then sin( `vec p` ^ `vec q)` is : |
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Answer» We know dot product of two perpendicular vectors is `0`. `:. (3vecp+vecq).(5vecp-3vecq) = 0` `=>15|vecp|^2-3|vecq|^2-4vecp*vecq = 0->(1)` Now, `(2vecp+vecq).(4vecp-2vecq) = 0` `=>8|vecp|^2-2|vecq|^2 = 0` `=>|vecq|^2 = 4|vecp|^2` `=>|vecq| = 2|vecp|` Putting value of `|vecq|` in (1), `15|vecp|^2-12|vecp|^2-4|vecp|*2|vecp|*costheta = 0` `3|vecp|^2-8|vecp|^2costheta = 0` `cos theta = 3/8` `:. sin theta = sqrt(1-(3/8)^2) = sqrt55/8` |
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