Book: Mathematics Part-II
Chapter: 11. Three Dimensional Geometry
Subject: Maths - Class 12th
Q. No. 1 of Exercise 11.2
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Show that the three lines with direction cosines 
are mutually perpendicular.
We know that
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then cos θ = |l1l2 + m1m2 + n1n2|
If two lines are perpendicular, then the angle between the two is θ = 90°
⇒ For perpendicular lines, | l1l2 + m1m2 + n1n2 | = cos 90° = 0, i.e.
| l1l2 + m1m2 + n1n2 | = 0
So, in order to check if the three lines are mutually perpendicular, we compute | l1l2 + m1m2 + n1n2 | for all the pairs of the three lines.
Now let the direction cosines of L1, L2 and L3 be l1, m1, n1; l2, m2, n2 and l3, m3, n3.
First, consider
⇒ 
⇒ L1⊥ L2 ……(i)
Next, consider
⇒ 
⇒ 
⇒ L2⊥ L3 …(ii)
Now, consider
⇒ 
⇒ 
⇒ L1⊥ L3 …(iii)
∴ By (i), (ii) and (iii), we have
L1, L2 and L3 are mutually perpendicular.
