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प्रश्न
The area of the triangle with vertices (1, 2, 0), (1, 0, 2) and (0, 3, 1) in sq. unit is ______.
पर्याय
`sqrt(5)`
`sqrt(7)`
`sqrt(6)`
`sqrt(3)`
उत्तर
The area of the triangle with vertices (1, 2, 0), (1, 0, 2) and (0, 3, 1) in sq. unit is `underlinebb(sqrt(6))`.
Explanation:
If A = (1, 2, 0), B = (1, 0 ,2) and C = (0, 3, 1),
then `vec(AB) = - 2hatj + 2hatk`, `vec(AC) = -i + j + k`
and the area of triangle ABC is `1/2 |vec(AB) xx vec(AC)|`
and `|vec(AB) xx vec(AC)| = |(hati, hatj, hatk),(0, -2, 2),(-1, 1, 1)| = -4hati - 2hatj - 2hatk`
`|vec(AB) xx vec(AC)|/2`
= `|-4i - 2j - 2k|/2`
= `sqrt(16 + 4 + 4)/2`
= `sqrt(24)/2`
= `(2sqrt(6))/2`
= `sqrt(6)` sq.units
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