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

The points

A (1, 1, 5), B (2, 2, 1), C (1, - 2, 2)

and

D (- 2, 1, 2)

are vertices of tetrahedron. Find the equation of the line through

A

perpendicular to face

B C D

and distance of

A

from this face. Also find the shortest distance between skew lines

A D

and

B C

.

Accepted Answer

We have,

$A (1, 1, 5), B (2, 2, 1), C (1, - 2, 2)$ and $D (- 2, 1, 2)$

Equation of plane $B C D$ is

$\Rightarrow |\begin{matrix} x - 2 & y - 2 & z - 1 \\ - 1 & - 4 & 1 \\ - 4 & - 1 & 1 \end{matrix}| = 0$

$\Rightarrow (x - 2) (- 3) - (y - 2) (3) + (z - 1) (- 15) = 0$

$\Rightarrow x - 2 + y - 2 + 5 z - 5 = 0$

$\Rightarrow x + y + 5 z = 9 . . . . (i)$

Equation of line through $A$ and perpendicular to $(i)$ is

$D (- 2, 1, 2)$

Distance of $A$ from plane $B C D$ is $= |\frac{1 + 1 + 25 - 9}{\sqrt{1 + 1 + 25}}| = \frac{18}{\sqrt{27}} = 2 \sqrt{3}$ units.

Equation of line $A D$ is $\vec{r} = (\hat{i} + \hat{j} + 5 \hat{k}) + λ (- 3 \hat{i} - 3 \hat{k})$

Equation of line $B C$ is $\vec{r} = (2 \hat{i} + 2 \hat{j} + \hat{k}) + μ (- \hat{i} - 4 \hat{j} + \hat{k})$

Now, ${\vec{a}}_{2} - {\vec{a}}_{1} = (\hat{i} + \hat{j} - 4 \hat{k})$

${\vec{b}}_{1} \times {\vec{b}}_{2} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 3 & 0 & - 3 \\ - 1 & - 4 & 1 \end{matrix}|$

$\Rightarrow {\vec{b}}_{1} \times {\vec{b}}_{2} = - 12 \hat{i} + 6 \hat{j} + 12 \hat{k}$

$\Rightarrow |{\vec{b}}_{1} \times {\vec{b}}_{2}| = \sqrt{144 + 144 + 36} = 18$

Hence, required shortest distance is

$= |\frac{(\vec{a_{2}} - \vec{a_{1}}) \cdot (\vec{b_{1}} \times \vec{b_{2}})}{|(\vec{b_{1}} \times \vec{b_{2}})|}|$

$= |\frac{(\hat{i} + \hat{j} - 4 \hat{k}) \cdot (- 12 \hat{i} + 6 \hat{j} + 12 \hat{k})}{18}|$

$= |\frac{- 12 + 6 - 48}{18}| = 3$ units.

The points $A (1, 1, 5), B (2, 2, 1), C (1, - 2, 2)$ and $D (- 2, 1, 2)$ are vertices of tetrahedron. Find the equation of the line through $A$ perpendicular to face $B C D$ and distance of $A$ from this face. Also find the shortest distance between skew lines $A D$ and $B C$ .

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

The points A1,1,5, B2,2,1, C1,-2,2 and D-2,1,2 are vertices of tetrahedron. Find the equation of the line through A perpendicular to face BCD and distance of A from this face. Also find the shortest distance between skew lines AD and BC.

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

The points $A (1, 1, 5), B (2, 2, 1), C (1, - 2, 2)$ and $D (- 2, 1, 2)$ are vertices of tetrahedron. Find the equation of the line through $A$ perpendicular to face $B C D$ and distance of $A$ from this face. Also find the shortest distance between skew lines $A D$ and $B C$ .