Name: The Shortest Distance Between Two Skew Lines
Uploaded: 2019-07-19
Duration: 9 min 21 s
Description: Skew lines are the lines that do not intersect and not parallel. The shortest distance between two skew lines lies along the line, which is perpendicular to...

Question

Find the length and the equation of the lines of shortest distances between the lines in the following case:

\frac{x - 8}{3} = \frac{y + 9}{- 16} = \frac{z - 10}{7}; \frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{- 5}

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Accepted Answer

Given Cartesian equations of the line are: $\frac{x - 8}{3} = \frac{y + 9}{- 16} = \frac{z - 10}{7} = k$ [say] and $\frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{- 5} = r$ [say]

Let any general point on the first line is $P$ i.e., $P (3 k + 8, - 16 k - 9, 7 k + 10)$

Let any general point on the second line is $Q$ i.e., $Q (3 r + 15, 8 r + 29, - 5 r + 5)$

Now, the direction ratios of $P Q$ are $3 r + 15 - 3 k - 8, 8 r + 29 + 16 k + 9, - 5 r + 5 - 7 k - 10$ i.e., $3 r - 3 k + 7, 8 r + 16 k + 38, - 5 r - 7 k - 5$

If $P Q$ be the line of shortest distance, then $P Q$ is perpendicular to both the lines.

So, $3 (3 r - 3 k + 7) - 16 (8 r + 16 k + 38) + 7 (- 5 r - 7 k - 5) = 0 and 3 (3 r - 3 k + 7) + 8 (8 r + 16 k + 38) - 5 (- 5 r - 7 k - 5) = 0$

$\Rightarrow 9 r - 9 k + 21 - 128 r - 256 k - 608 - 35 r - 49 k - 35 = 0 and 9 r - 9 k + 21 + 64 r + 128 k + 304 + 25 r + 35 k + 25) = 0$

$\Rightarrow - 154 r - 314 k - 622 = 0 and 98 r + 154 k + 350 = 0$

$\Rightarrow 77 r + 157 k + 311 = 0 \to (1) and 49 r + 77 k + 175 = 0 \to (2)$

Solving $(1)$ and $(2)$ , then we get: $k = - 1$ and $r = - 2$ .

Thus, the coordinates of $P$ are $P (- 3 + 8, 16 - 9, - 7 + 10)$ i.e., $P (5, 7, 3)$

And, the coordinates of $Q$ are $Q (- 6 + 15, - 16 + 29, 10 + 5)$ i.e., $Q (9, 13, 15)$

Thus, length of the shortest distance $= P Q = \sqrt{(9 - 5)^{2} + (13 - 7)^{2} + {(15 - 3)}^{2}} = \sqrt{16 + 36 + 144} = \sqrt{196} = 14$ units.

Equations of the line of shortest distance is the equation of the line passing through the points $P (5, 7, 3)$ and $Q (9, 13, 15)$ i.e., $\frac{x - 5}{9 - 5} = \frac{y - 7}{13 - 7} = \frac{z - 3}{15 - 3}$ or $\frac{x - 5}{4} = \frac{y - 7}{6} = \frac{z - 3}{12}$ or $\frac{x - 5}{2} = \frac{y - 7}{3} = \frac{z - 3}{6}$

Hence, the equation of the line of shortest distance is $\frac{x - 5}{2} = \frac{y - 7}{3} = \frac{z - 3}{6}$ .

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