Find the shortest distance between the parallel lines $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(2\hat i+4\hat j+5\hat k)+\mu(4\hat i+6\hat j+8\hat k)$.

Find the shortest distance between the parallel lines vec r=(hat i+2hat j+3hat k)+λ(2hat i+3hat j+4hat k) and

Exercise: Skew Lines and Shortest Dis... · Q27

\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)

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$\vec b_2=4\hat i+6\hat j+8\hat k=2(2\hat i+3\hat j+4\hat k)=2\vec b_1$ , so the lines are parallel; take the common direction $\vec b=2\hat i+3\hat j+4\hat k$ , with $|\vec b|=\sqrt{4+9+16}=\sqrt{29}$ .

Joining vector: $\vec a_2-\vec a_1=(2-1)\hat i+(4-2)\hat j+(5-3)\hat k=\hat i+2\hat j+2\hat k$ . …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2016Set ANNUAL1 markMCQ
$\dfrac{x-3}{4}=\dfrac{y-1}{2}=\dfrac{z-5}{-3}$
(a) $3$
(b) $2$
(c) $1$
(d) $0$
›Reveal solutionSolution
Using the shortest-distance-between-parallel-lines formula, the perpendicular distance works out exactly to $3$ .

Line 1: $\dfrac{x-3}{4}=\dfrac{y-1}{2}=\dfrac{z-5}{-3}$ passes through $A_1(3,1,5)$ with direction $\vec b=4\vec i+2\vec j-3\vec k$ .

Line 2: $\dfrac{x-1}{4}=\dfrac{y-2}{2}=\dfrac{z-3}{-3}$ passes through $A_2(1,2,3)$ with the same direction $\vec b$ , confirming the lines are parallel.

Vector joining the two points: $\vec{A_1A_2}=(1-3)\vec i+(2-1)\vec j+(3-5)\vec k=-2\vec i+\vec j-2\vec k$ .

Shortest distance between parallel lines: $d=\dfrac{\left|\vec{A_1A_2}\times\vec b\right|}{|\vec b|}$ .

Compute the cross product: $\vec{A_1A_2}\times\vec b=\begin{vmatrix}\vec i&\vec j&\vec k\\-2&1&-2\\4&2&-3\end{vmatrix}=(1)(1\vec i)+\ldots$ Component-wise: $i$ -comp $=(1)(-3)-(-2)(2)=-3+4=1$ ; $j$ -comp $=-[(-2)(-3)-(-2)(4)]=-[6+8]=-14$ ; $k$ -comp $=(-2)(2)-(1)(4)=-4-4=-8$ . So the cross product is $(1,-14,-8)$ . …
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Q.Find the shortest distance between the parallel lines $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(2\hat i+4\hat j+5\hat k)+\mu(4\hat i+6\hat j+8\hat k)$ .

Concept understanding — Skew Lines and Shortest Distance

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