Question

A line passed through the point $A(6,2,2)$ and is parallel to the vector $(1,-2,2)$. Another line passes through the point $B(-4,0,-1)$ and is parallel to the vector $(3,-2,-2)$. The shortest distance between these two lines is

• A. $3$
• B. $6$
• C. $9$
• D. $12$

Answer 1

The correct option is C $9$

Line passing through point and parallel to vector is given by

$\overrightarrow{r}=(6,2,2)+t(1,-2,2)\overline{r}$

$=(-7,0-1)+5(3,-2,-2)$

$\overrightarrow{b}\times \overrightarrow{d}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 2\\ 3& -2& -2\end{array}\right|=8\hat{i}+8\hat{j}+4\hat{k}$

$(\overline{a}-\overline{c})=(10,2,3)$

Shortest Distance is
$=\frac{\left|\right(\overline{a}-\overrightarrow{c}).\overline{b}\times \overline{d}|}{|\overrightarrow{b}\times \overrightarrow{d}|}=\frac{80+16+12}{12}=9$