Question

If l,m,n and a,b,c are direction cosines of two lines then

• A. they are parallel when la + mb + nc = 0
• B. they are perpendicular when \(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}\)
• C. the direction ratios of the bisectors of the angles between l±a, m±b, n±c
• D. the direction ratios of the bisectors of the angles between l±a, m±b, n±c

Answer 1

The Correct Option is C

To solve the problem, we need to evaluate the given options about the direction cosines of two lines and identify the correct statement.

1. Understand Direction Cosines Properties:
Let $l, m, n$ and $a, b, c$ be the direction cosines of two lines.
Two lines are parallel if their direction cosines are proportional: $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$.
Two lines are perpendicular if the dot product of their direction cosines is zero: $la + mb + nc = 0$.
The direction ratios of the angle bisectors between the lines are $l \pm a, m \pm b, n \pm c$.

2. Evaluate the Given Options:
Option 1 states the lines are parallel when $la + mb + nc = 0$.
This is incorrect, as $la + mb + nc = 0$ indicates perpendicularity, not parallelism.
Option 2 states the lines are perpendicular when $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$.
This is incorrect, as $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$ indicates parallelism, not perpendicularity.
Option 3 states the direction ratios of the angle bisectors are $l \pm a, m \pm b, n \pm c$.
This is correct, as it matches the standard formula for the direction ratios of angle bisectors.

3. Conclusion:
Options 1 and 2 are incorrect, but option 3 is correct.

Final Answer:
The direction ratios of the angle bisectors are $l \pm a, m \pm b, n \pm c$.