If A(3, 1, –1), , C(2, 2, 1) and are the vertices of a quadrilateral ABCD, then its area is [2024]

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Solution

Q.
If A(3, 1, –1), $B (\frac{5}{3}, \frac{7}{3}, \frac{1}{3})$ , C(2, 2, 1) and $D (\frac{10}{3}, \frac{2}{3}, \frac{- 1}{3})$ are the vertices of a quadrilateral ABCD, then its area is [2024]

1 $\frac{5 \sqrt{2}}{3}$

2 $\frac{4 \sqrt{2}}{3}$

3 $2 \sqrt{2}$

4 $\frac{2 \sqrt{2}}{3}$

Ans.
(2)

Gicen, A(3, 1, –1), $B (\frac{5}{3}, \frac{7}{3}, \frac{1}{3})$ , C(2, 2, 1) and $D (\frac{10}{3}, \frac{2}{3}, \frac{- 1}{3})$ are vertices of quadrilateral ABCD

$\vec{A C} = (2 - 3) \hat{i} + (2 - 1) \hat{j} + (1 + 1) \hat{k} = - \hat{i} + \hat{j} + 2 \hat{k}$

$\vec{B D} = (\frac{10}{3} - \frac{5}{3}) \hat{i} + (\frac{2}{3} - \frac{7}{3}) \hat{j} + (\frac{- 1}{3} - \frac{1}{3}) \hat{k} = \frac{5}{3} \hat{i} - \frac{5}{3} \hat{j} - \frac{2}{3} \hat{k}$

Area of quadrilateral ABCD = $\frac{1}{2}$ $| \vec{A C} \times \vec{B D} |$

$= \frac{1}{2}$ $| | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 1 & 1 & 2 \\ 5 / 3 & - 5 / 3 & - 2 / 3 \end{matrix} | |$ $= \frac{1}{2}$ $| \hat{i} (\frac{8}{3}) - \hat{j} (\frac{- 8}{3}) + \hat{k} (0) |$

$= \frac{1}{2} \sqrt{\frac{64}{9} + \frac{64}{9}} = \frac{4 \sqrt{2}}{3} sq . units$ .

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