Let ` vec a , vec ba n d vec c` be three verctors such that ` vec a!=0,| vec a|=| vec c|=1,| vec b|=4` and `| vec bxx vec c|=sqrt(15)dot` If ` vec b-2 vec c=lambda vec a ,` then find the value of `lambda`

Explanation

Let angle between `overset(1)(b)` and `overset(1)( c )` is `alpha`
Given `
|overset(1)(b) xx overset(1) ( c ) | = sqrt(15) rArr | overset( 1)(b) ||overset(1) (c ) | sin alpha = sqrt( 15) rArr sin alpha = ( sqrt( 15))/( 4)`
`:. cos alpha = sqrt( 1- sin^(2) alpha ) = sqrt( 1- ( 15)/( 16)) = (1)/(4)`
Q ` overset(1)(b) - 2 overset(1) ( c ) = lambda overset(1) ( a) `
`rArr ( overset( 1) ( b ) - 2 overset( 1) (c )) = lambda^(2) ( overset(1) ( a) ) ^(2) rArr overset(r_(2))(b)+4overset(r_(2))(c) - 4overset(r)(b).overset(r )(c ) = lambda^(2) overset(r_(2))(a)`
`rArr 16+4 xx 1 - 4 (|overset(1)(b)||overset(1)(c)|cos alpha ) = lambda^(2) . 1^(2)`
`rArr 20 - 4 = lambda^(2) rArr lambda = +-4 rArr | lambda | = 4 `