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Question
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.
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Solution
Two lines whose direction cosines are l1, m1, n1 and l2, m2, n2 respectively will be mutually perpendicular.
If, l1l2 + m1m2 + n1n2 = 0
(i) Lines with direction cosines `12/13, (-3)/13, (-4)/13; 4/13, 12/13, 3/13`
Here the values of l1, m1, n1 are `2/13, (-3)/13, (-4)/13` respectively
And the values of l2, m2, n2 are `4/13, 12/13, 3/13` respectively.
Now, l1l2 + m1m2 + n1n2
= `12/13 xx 4/13 + ((-3)/13) (12/13) + ((-4)/13) (3/13)`
`= 48/169 - 36/169 - 12/169`
= 0
⇒ Hence, both the lines are perpendicular to each other.
(ii) Lines with direction cosines `4/13, 12/13, 3/13; 3/13, (-4)/13, 12/13`
Now, l1l2 + m1m2 + n1n2
= `4/13 xx 3/13 + 12/13 ((-4)/13) + 3/13 xx (12/13)`
`= 12/169 - 48/169 + 36/169`
= `(12 - 48 + 36)/169`
= 0
⇒ These two lines are perpendicular to each other.
(iii) Lines with direction cosines `3/13, (-4)/13, 12/13; 12/13, (-3)/13, (-4)/13`
Now, l1l2 + m1m2 + n1n2
= `3/13 xx 12/13 + ((-4)/13) ((-3)/13) + 12/13 xx ((-4)/13)`
`= 36/169 + 12/169 - 48/169`
= `(36 + 12 - 48)/169`
= 0
⇒ Both lines are perpendicular to each other.
Therefore, the given lines are perpendicular to each other.
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