Question

The distance between two parallel lines can be taken out by the formula

• A. $d=\left|\frac{b\cdot \left(a_2-a_1\right)}{|b|}\right|$
• B. $d=\left|\frac{b\cdot \left(a_2-a_1\right)}{|a_2-a_1|}\right|$
• C. $d=\left|\frac{b\times \left(a_2-a_1\right)}{|a_2-a_1|}\right|$
• D. $d=\left|\frac{b\times \left(a_2-a_1\right)}{|b|}\right|$

Answer 1

Answer: (D)

If two lines $l_1$ and $l_2$ are parallel, then they are coplanar.
Let the lines be given by
$r=a_1+\lambda b$ ...(i)
and $r=a_2+\mu b$....(ii)
where $a_1$ is the position vector of a point $S$ on $l_1$ and $a_2$ is the position vector of a point $T$ on $I_2$.
As $I_1,I_2$ are coplanar, if the foot of the perpendicular from $T$ on the line $l_1$ is $P$, then the distance between the lines, $l_1$ and $I_2=|TP|$.

Let $\theta$ be the angle between the vector ST and $b$.
Then, $b\times ST=(|b||ST|\sin \theta )n$ where $\hat{n}$ is the unit vector perpendicular to the plane of the line $I_1$ and $I_2$
But $ST=a_2-a_1$
Therefore, from Eq. (iii), we get
$b\times \left(a_2-a_1\right)=|b|PT\hat{n}(\because PT-ST\sin \theta )$
$\text{ i.e., }\left|b\times \left(a_2-a_1\right)\right|=|b|PT\cdot 1(as|\hat{n}|=1)$
Hence, the distance between the given parallel lines is
$d=|PT|=\left|\frac{b\times \left(a_2-a_1\right)}{|b|}\right|$