JEE Main 2024 — 5 April, Shift 1 — Question 18

$\mathrm{A}(1,-1,2)$

$\mathrm{B}(5,7,-6)$

$\mathrm{C}(3,4,-10)$

$\mathrm{D}(-1,-4,-2)$

Area $=\frac{1}{2}|\overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{BD}}|=\frac{1}{2}|(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-12 \hat{\mathrm{k}}) \times(6 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})|$

$=\frac{1}{2}|112 \hat{\mathrm{i}}-64 \hat{\mathrm{j}}-8 \hat{\mathrm{k}}|$

$=4|14 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}-\hat{\mathrm{k}}|$

$=4 \sqrt{196+64+1}$

$=4 \sqrt{261}$

$=12 \sqrt{29}$