Prove that two lines whose direction cosines are given by relation, pl + qm + rn =0 and a(l^2) + b(m^2) + c(n^2) =0 are perpendicular or parallel according as p^2(b+c) + q^2(c+a) + r^2(a+b) =0 or p^2/a + q^2/b r^2/c =0.

To determine the relationship between two lines based on their direction cosines, we need to analyze the given equations and their implications. The direction cosines of a line are typically represented by (l, m, n), which correspond to the angles that the line makes with the coordinate axes. Let's break down the problem step by step.

Understanding the Direction Cosines

The direction cosines of a line are defined as:

• l = cos(α), where α is the angle with the x-axis
• m = cos(β), where β is the angle with the y-axis
• n = cos(γ), where γ is the angle with the z-axis

For two lines to be either parallel or perpendicular, their direction cosines must satisfy certain conditions.

Analyzing the Given Relations

We have two equations:

• pl + qm + rn = 0
• a(l²) + b(m²) + c(n²) = 0

These equations represent conditions that the direction cosines of the lines must satisfy. The first equation indicates a linear combination of the direction cosines, while the second involves their squares, weighted by constants a, b, and c.

Conditions for Perpendicularity and Parallelism

To prove that the lines are perpendicular or parallel, we need to analyze the expression:

• p²(b + c) + q²(c + a) + r²(a + b) = 0
• p²/a + q²/b + r²/c = 0

Case 1: Perpendicular Lines

For the lines to be perpendicular, the first condition must hold:

If p²(b + c) + q²(c + a) + r²(a + b) = 0, it implies that the weighted sums of the squares of the direction cosines are balanced in such a way that the lines cannot intersect at an angle other than 90 degrees. This means that the direction cosines are orthogonal.

Case 2: Parallel Lines

On the other hand, if p²/a + q²/b + r²/c = 0, this indicates that the ratios of the squares of the direction cosines to their respective coefficients are equal. This equality suggests that the lines maintain a constant direction, hence they are parallel.

Conclusion

In summary, the relationships between the direction cosines and the conditions provided allow us to conclude that:

• If the first condition holds, the lines are perpendicular.
• If the second condition holds, the lines are parallel.

This analysis demonstrates how the geometric properties of lines in three-dimensional space can be expressed algebraically through their direction cosines. Understanding these relationships is crucial in fields such as physics and engineering, where the orientation of lines and vectors plays a significant role.