Question -
Answer -
Let us consider thedirection cosines of L1, L2 and L3 bel1, m1, n1; l2, m2, n2 andl3, m3, n3.
We know that
If l1, m1,n1 and l2, m2, n2 are thedirection cosines of two lines;
And θ is the acuteangle between the two lines;
Then cos θ = |l1l2 +m1m2 + n1n2|
If two lines areperpendicular, then the angle between the two is θ = 90°
For perpendicularlines, | l1l2 + m1m2 + n1n2 |= cos 90° = 0, i.e. | l1l2 + m1m2 +n1n2 | = 0
So, in order to checkif the three lines are mutually perpendicular, we compute | l1l2 +m1m2 + n1n2 | for all thepairs of the three lines.
Firstly let uscompute, | l1l2 + m1m2 +n1n2 |
So, L1⊥ L2 …… (1)
Similarly,
Let us compute, | l2l3 +m2m3 + n2n3 |
So, L2⊥ L3 ….. (2)
Similarly,
Let us compute, | l3l1 +m3m1 + n3n1 |
So, L1⊥ L3 ….. (3)
∴ By (1), (2) and(3), the lines are perpendicular.
L1, L2 andL3 are mutually perpendicular.