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JEE Main 20246 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
6 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Straight Lines

JEE Main 2024 · 6 April · Morning Shift · Mathematics · Straight Lines

If A(3,1,-1), B((5)/(3), (7)/(3), (1)/(3)), C(2,2,1) and...

If A(3,1,−1),B(53,73,13),C(2,2,1)A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1) and D(103,23,−13)D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right) are the vertices of a quadrilateral ABCDA B C D, then its area isIf A(3,1,-1), B((5)/(3), (7)/(3), (1)/(3)), C(2,2,1) and D((10)/(3), (2)/(3), (-1)/(3)) are the vertices of a

Options

  1. A223\frac{2 \sqrt{2}}{3}
  2. B523\frac{5 \sqrt{2}}{3}
  3. C222 \sqrt{2}
  4. D423\frac{4 \sqrt{2}}{3}

Correct answer

D. 423\frac{4 \sqrt{2}}{3}

Step-by-step solution

If A(3,1,-1), B((5)/(3), (7)/(3), (1)/(3)), C(2,2,1) and D((10)/(3), (2)/(3), (-1)/(3)) are the vertices of a
Area =12\frac{1}{2}|\overline{\mathrm{BD}}\times \overline{AC\mathrm{AC}}| BD‾=53i^−53j^−23k^AC‾=i^−j^−2k^\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}
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