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Question
Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is ______.
Options
`1/2`
1
2
4
Solution
Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is 2.
Explanation:
`vec(OA) = -hati + 1/2hatj + 4hatk,`
` vec(OB) = hati 1/2hatj + 4hatk,`
` vec(OC) = -hati - 1/2hatj + 4hatk`
`vec(AB) = (1 + 1)hati + (1/2 - 1/2)hatj + (4 - 4)hatk `
`= 2hati`
`vec(BC) = (1 - 1)hati + (-1/2 - 1/2)hatj + (4 - 4)hatk`
` = -hatj`
`vec(AB) xx vec(BC) = |(hati, hatj, hatk), (2, 0, 0), (0, -1, 0)| = -2hatk`
`|vec(AB) xx vec(BC)| = sqrt((-2)^2) = 2`
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