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Vector Algebra

Question
CBSEENMA12034171
Wired Faculty App

Find a unit vector perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top if  straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top space space and space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 3 space straight k with hat on top

Solution

Here,  straight a with rightwards arrow on top space equals 4 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 3 space straight k with hat on top
therefore space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top 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 4 3 2 row 2 5 cell negative 3 end cell end table close vertical bar space equals space left parenthesis negative 9 minus 10 right parenthesis space straight i with hat on top space minus space left parenthesis negative 12 minus 4 right parenthesis straight j with hat on top space plus space left parenthesis 20 minus 6 right parenthesis space straight k with hat on top space space space space equals space minus 19 space straight i with hat on top space plus space 16 space straight j with hat on top space plus 14 space straight k with hat on top
which is a vector perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.
open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals square root of left parenthesis negative 19 right parenthesis squared plus left parenthesis 16 right parenthesis squared plus left parenthesis 14 right parenthesis squared end root space equals square root of 361 plus 256 plus 196 end root space equals square root of 813
therefore space space space a unit vector perpendicular to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space equals space fraction numerator straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar end fraction
                                                                equals space fraction numerator 1 over denominator square root of 813 end fraction left parenthesis negative 19 space straight i with hat on top space plus space 16 space straight j with hat on top space plus space 14 space straight k with hat on top right parenthesis

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