Explanation:
D Let, \(\mathrm{P}=(5,7,3), \mathrm{Q}=(9,13,15)\) and \(\mathrm{R}=(12\), 21,10 ).
Let \(A\) be the foot of perpendicular from \(P\) to \(Q R\).
Also, \(\mathrm{A}\) divides \(\mathrm{QR}\) in \(\lambda: 1\)
\(\therefore A=\left(\frac{9+12 \lambda}{1+\lambda}, \frac{13+21 \lambda}{1+\lambda}, \frac{15+10 \lambda}{1+\lambda}\right)\{\) section formula \(\}\)
Direction ratio of \(\mathrm{PA}-\)
\(\left(\frac{9+12 \lambda}{1+\lambda}-5, \frac{13+21 \lambda}{1+\lambda}-7, \frac{15+10 \lambda}{1+\lambda}-3\right)\)
Direction ratio of \(\mathrm{QR}\)..
\((12-9,21-13,10-15)=(3,8,-5) \text {. }\)
Now, \(\overrightarrow{\mathrm{PA}} \perp \overrightarrow{\mathrm{QR}}\) i.e. \(\overrightarrow{\mathrm{PA}} \cdot \overrightarrow{\mathrm{QR}}=0\)
\(\Rightarrow\)
\(3\left[\frac{9+12 \lambda}{1+\lambda}-5\right]+8\left[\frac{13+21 \lambda}{1+\lambda}-7\right]-5\left[\frac{15+10 \lambda}{1+\lambda}-3\right]=0\)
\(\Rightarrow \lambda=0\)
\(\therefore\) Foot of perpendicular
\(A=(9,13,15) \text {. }\)