JEE Main 2024 — 1 February, Shift 2 — Question 9

$\mathrm{A}(1,3,2) ; \mathrm{B}(-2,8,0) ; \mathrm{C}(3,6,7)$

$\overrightarrow{\mathrm{AC}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$

$\mathrm{AB}=\sqrt{9+25+4}=\sqrt{38}$

$\mathrm{AC}=\sqrt{4+9+25}=\sqrt{38}$

$\overrightarrow{\mathrm{AD}}=-\frac{1}{2} \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\frac{3}{2} \hat{\mathrm{k}}=-\frac{1}{2}(\hat{\mathrm{i}}+8 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$

Length of projection of

$\overrightarrow{\mathrm{AD}}$ on $\overrightarrow{\mathrm{AC}}$

$=\left|\frac{\overrightarrow{\mathrm{AD}} \cdot \overrightarrow{\mathrm{AC}}}{|\overrightarrow{\mathrm{AC}}|}\right|=\frac{37}{2 \sqrt{38}}$