Show that the three lines with direction cosines `(12)/(13),(-3)/(13),(-4)/(13),4/(13),(12)/(13),3/(13);3/(13),(-4)/(13),(12)/(13)`are mutually perpendicular.

Explanation

(i) For first two lines ` l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)`
`= 12/13 xx 4/13 + (-3/13) xx 12/13 + (-4/13) xx 3/13`
`= 48/169- 36/169 - 12/169 = 0`
Therefore , two lines are perpendicular
(ii) For second and third line, `l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)`
` = 4/13 xx 3/12 + 12/13 xx ((-4)/(13)) + 3/13 xx 12/13`
`=12/169-48/1639+36/169=0`
Therefore, two lines are perpendicular
(iii) For third and first line
`l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)`
`= 3/13 xx 12/13+((-4)/(13)) xx ((-3)/(13)) + (12)/(13) xx ((-4)/(13))`
`= 36/169+12/169-48/169=36/169-36/169=0`
Therefore, two lines are perpendicular.
Hence, the lines are mutually perpendicular.