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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
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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