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Question
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
Options
9 sq.units
18 sq.units
27 sq.units
81 sq.units
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Solution
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to 9 sq.units.
Explanation:
We have, `vec(AB) = (2 - 0)hati + (3 - 4)hatj + (-1 -1)hatk = 2hati - hatj - 2hatk`
`vec(BC) = (4 - 2)hati + (5 - 3)hatj + (0 + 1)hatk = 2hati + 2hatj + hatk`
`vec(CD) = (2 - 4)hati + (6 - 5)hatj + (2 - 0)hatk = -2hati + hatj + 2hatk`
`vec(DA) = (0 - 2)hati + (4 - 6)hatj + (1 - 2)hatk = -2hati - 2hatj - hatk`
∴ Area of quadrilateral ABCD = `|vec(AB) xx vec(BC)|`
= `|(hati, hatj, hatk),(2, -1, -2),(2, 2, 1)|`
= `|hati(-1 + 4) - hatj(2 - 4) + hatk(4 + 2)|`
= `|3hati - 6hatj + 6hatk|`
= `sqrt(9 + 36 + 36)`
= 9 sq.units
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