`veca = 3i + j` व `vecb = 2i - j - 5k ` के लम्बवत एकांक सदिश होगा-

Options

A. `pm ((i - 3 j + k))/sqrt(11) `
B. `pm (3i + j)/sqrt11`
C. `pm((2i - j - 5k))/sqrt30 `
D. इनमें से कोई नहीं

Answer

A

Explanation

लम्बवत दिशा में एकांक सदिश
`hatn = pm (a xx b)/(|a||b|sin theta) `
`a xx b = |(i,j,k),(3,1,0),(2,-1,-5)|`
` = i (-5 - 0) - j (-15 - 0) + k (-3 -2) `
` = - 5i + 15j - 5 k`
`|a| = sqrt(3^(2) + 1^(2)) = sqrt(10)`
`|b| = sqrt((2)^(2) + (-1)^(2) + (-5)^(2)) = sqrt(30) `
` cos theta = (a *b)/(|a||b|) = (6-1)/(sqrt 10 sqrt 30) = 5/(10 sqrt3) = 1/(2 sqrt 3) `
` :. sin theta = sqrt(1 - cos^(2) theta) `
` = sqrt(1 - 1/12) = sqrt(11/12) = sqrt11/(2sqrt3) `
`hat n = pm (-5 i + 15 j - 5k)/(sqrt10 sqrt30 sqrt11/(2sqrt3)) = pm (-5(i - 3 j + k))/(10 sqrt3 sqrt 11//2 sqrt3) `
` = pm {(-(i - 3j + k))/sqrt11} = pm (i - 3j + k)/sqrt 11 `