The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(?2, ?3, 5) and D(1, ?6, ?7) is equal to [2023]

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Solution

Q.
The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to [2023]

1 $8 \sqrt{38}$

2 $48$

3 $54$

4 $9 \sqrt{38}$

Ans.
(1)

$The area of quadrilateral A B C D is equal to \frac{1}{2}$ $| \vec{A C} \times \vec{B D} |$ .

Now, $\vec{A C} = (- 2 \hat{i} - 3 \hat{j} + 5 \hat{k}) - (2 \hat{i} + \hat{j} + \hat{k}) = - 4 \hat{i} - 4 \hat{j} + 4 \hat{k}$

and $\vec{B D} = (\hat{i} - 6 \hat{j} - 7 \hat{k}) - (\hat{i} + 2 \hat{j} + 5 \hat{k}) = - 8 \hat{j} - 12 \hat{k}$

So, $\frac{1}{2}$ $| \vec{A C} \times \vec{B D} |$ $= \frac{1}{2}$ $| | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 4 & - 4 & 4 \\ 0 & - 8 & - 12 \end{matrix} | |$

$= \frac{1}{2}$ $| 80 \hat{i} - 48 \hat{j} + 32 \hat{k} |$ $= \frac{1}{2} \sqrt{9728} = 8 \sqrt{38}$

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