The area of the parallelogram formed by the points $(1, 1, 1), (- 1, 5, 5), (2, 2, 5)$ is

81

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9

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36

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18

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Solution

The correct option is C $18$
$A = (1, 1, 1), B = (- 1, 5, 5), C = (2, 2, 5)$
$- - \to A B = (- i + 5 j + 5 k) - (i + j + k)$
$= (- 2 j + 4 j + 4 k)$
$- - \to A C = (2 i + 2 j + 5 k) - (i + j + k)$
$= (i + j + 4 k)$
$A r e a o f p a r a l l e l o g r a m = m o d o f - - \to A B \times - - \to A C$
$= m o d o f ∣ ∣ ∣ ∣ \begin{matrix} i & j & k - 2 & 4 & 4 1 & 1 & 4 \end{matrix} ∣ ∣ ∣ ∣$
$= | (16 - 4) i - (- 8 - 4) j + (- 2 - 4) k |$
$= | 12 i + 12 j - k |$
$= \sqrt{144 + 144 + 36}$
$= \sqrt{324} = 18$

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