Question

Find the length and foot of the perpendicular segment from $P(1,2,-3)$ to the line $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}$

• A. $\sqrt{10}$
• B. $\sqrt{11}$
• C. $\sqrt{7}$
• D. None of these

Answer 1

Answer: (B)

$\sqrt{11}$

Let the direction ratio of perpendicular segment be $(a,b,c)$

$\therefore \cos \theta =\frac{-2\times 2+1\times (-2)+(-1)\times 3}{\sqrt{(-2{)}^2+(1{)}^2+(3{)}^2}\sqrt{(2{)}^2+(2{)}^2+(-1{)}^2}}$

$\therefore \cos \theta =\frac{-9}{\sqrt{14}\sqrt{9}}$

$\cos \theta =\frac{-3}{\sqrt{14}}$

$\Rightarrow \sin \theta =\frac{\sqrt{11}}{\sqrt{14}}$

$\therefore$ Length of perpendicular $PA=PB\sin \theta$

$=\sqrt{(-1-1{)}^2+(3-2{)}^2+(0+3{)}^2}\times \frac{\sqrt{11}}{\sqrt{14}}$

$=\sqrt{11}$