Question

The lines $\overrightarrow{r}$= i – j + l(2i + k) and $\overrightarrow{r}$= (2i – j) + m(i + j – k) intersect for

• A. l = 1, m = 1
• B. l = 2, m = 3
• C. All values of l and m
• D. No value of l and m

Answer 1

Answer: (D)

No value of l and m

Answer 2

Sol. The given lines intersect, if the shortest distance between the lines is zero.

We know that the shortest distance between the lines r = a₁ + (l${\overrightarrow{b}}_{1}$) and r = a₂ + lb₂ is

$\frac{|(a_{1} - a_{2}).b_{1} \times b_{2}|}{|b_{1} \times b_{2}|}$

So the shortest distance between the given lines is zero if

(i – j – (2i – j) . (2i + k) × (i + j – k) = 0

L.H.S. = $\left| \begin{matrix}

• 1 & 0 & 0 \ 2 & 0 & 1 \ 1 & 1 & - 1 \end{matrix} \right|$ = 1 ¹ 0

Hence the given lines do not intersect.