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If overset(to)(A) = 2hat(i) + hat(k) , o...

If `overset(to)(A) = 2hat(i) + hat(k) , overset(to)(B) = hat(i) + hat(j) +hat(k) " and " overset(to) (C ) = 4hat(i) - 3hat(j) +7hat(k)`
Determine a vector `overset(to)(R ) " satisfying " overset(to)(R ) xx overset(to)( B) = overset(to)( C ) xx overset(to)( B) " and " overset(to)(R ) " ." overset(to)(A) = 0`

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The correct Answer is:
`-(hat(i) - 8hat(j) + 2hat(k)`
Let `vec(R ) = xhat(i) + yhat(j) +zhat(k)`
`:., vec(R ) xx vec(B) = vec(C ) xx vec(B )`
`rArr |{:(hat(i),,hat(j),,hat(k)),(x,,y,,z),(1,,1,,1):}|= |{:(hat(i),,hat(j),,hat(k)),(4,,-3,,7),(1,,1,,1):}|`
`rArr (y-z) hat(i) -(x -z) hat(j) +(x-y) hat(k) =- 10hat(i) -3hat(j) +7hat(k) `
`rArr y-z =- 10, z =- 3 , x -y =7`
and ` vec(R ) ". " vec(A ) =0 rArr 2x + z=0`
On solving above equations x=- 1 ,y=-8 and z=2
`:. vec(R ) =- hat(i) -8hat(j) +2hat(k)`
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