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Abhinav Suraj Cheema
Asked: 2022-11-08T18:06:04+05:30 In:

The position vectors of thepoint `A ,B ,Ca n dDa r e3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k ,- hat i+ hat j+2 hat k`and 4` hat i+5 hat j+lambda hat k ,`respectively. Ifthe points `A ,B ,Ca n dD`lie on a plane, find the value of `lambdadot`

The position vectors of thepoint `A ,B ,Ca n dDa r e3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k ,- hat i+ hat j+2 hat k`and 4` hat i+5 hat j+lambda hat k ,`respectively. Ifthe points `A ,B ,Ca n dD`lie on a plane, find the value of `lambdadot`
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  1. Correct Answer – `(146)/(17)`
    Here `vec(AB) =- hat(i) +5hat(j) – 3hat(k)`
    `vec(AC) =- 4hat(i) + 3hat(j) +3hat(k) ” and ” vec(AD) =hat(i) +7hat(j) + (lambda + 1) hat(k)`
    we know that A,B,C,D lie in a plane if `vec(AB) , vec(AC) , vec(AD)` are coplanar i.e,
    `[vec(AB) vec(AC) vec(AD)]=0 rArr |{:(-1,,5,,-3),(-4,,3,,3),(1,,7,,lambda+1):}|=0`
    `rArr -1 (3lambda +3 -21) -5 (-4lambda -4-3) -3 (-28 -3) =0`
    `rArr -17lambda + 146 =0`
    `:. lambda =(146)/(17)`

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