If $A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)$ and $D\left(\frac{10}{3},…

If $A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)$ and $D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral $A B C D$, then its area is
  1. $\frac{2 \sqrt{2}}{3}$
  2. $\frac{5 \sqrt{2}}{3}$
  3. $2 \sqrt{2}$
  4. $\frac{4 \sqrt{2}}{3}$

Solution


Area $=\frac{1}{2}|\overline{\mathrm{BD}} \times \overline{\mathrm{AC}}|$ $\begin{aligned} & \overline{\mathrm{BD}}=\frac{5}{3} \hat{\mathrm{i}}-\frac{5}{3} \hat{\mathrm{j}}-\frac{2}{3} \hat{\mathrm{k}} \\ & \overline{\mathrm{AC}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\end{aligned}$

Asked in: JEE Main 2024 (06 Apr Shift 1)

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