The shortest distance between the straight lines through the points $A_{1} = (6,2,2)\$and$\ A_{2} = ( - 4,0, - 1)$ in the directions of $(1, - 2,2)$ and $(3, - 2, - 2)$ is

Question

Dr. Abubacker · Accepted Answer

(d)
$Here,\ \overrightarrow{\mathbf{a}_{\mathbf{1}}} = 6\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}},\ \overrightarrow{\mathbf{a}_{\mathbf{2}}} = - 4\widehat{\dot{\mathbf{i}}} + 0\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}},\$
$\overrightarrow{\mathbf{b}_{\mathbf{1}}} = \widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}\ \$and$\ \ \overrightarrow{\mathbf{b}_{\mathbf{2}}} = 3\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}}\$
∴Shortest distance
$= \left| \frac{\left( \overrightarrow{\mathbf{a}_{\mathbf{2}}} - \overrightarrow{\mathbf{a}_{\mathbf{1}}} ight) \bullet (\overrightarrow{\mathbf{b}_{\mathbf{1}}} 	imes \overrightarrow{\mathbf{b}_{\mathbf{2}}})}{\left| \overrightarrow{\mathbf{b}_{\mathbf{1}}} 	imes \overrightarrow{\mathbf{b}_{\mathbf{2}}} ight|} ight|$
$= \left| \frac{\left( - 10\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}} ight) \bullet (8\widehat{\dot{\mathbf{i}}} + 8\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}})}{\sqrt{64 + 64 + 16}} ight|$
$= \left| - \frac{108}{12} ight| = 9$

Q. The shortest distance between the straight lines through the points $A_{1} = (6,2,2)\$and$\ A_{2} = ( - 4,0, - 1)$ in the directions of $(1, - 2,2)$ and $(3, - 2, - 2)$ is

Explanation

Reviewed by Dr. Abubacker

Want more practice?