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 this problem, we first need to understand the terms involved and their geometrical implications:

In three-dimensional geometry, direction cosines are cosines of angles between a line and the coordinate axes. For a line with direction cosines \(l\), \(m\), \(n\), these angles are formed with the x-axis, y-axis, and z-axis, respectively.

Let us examine each given scenario:

1. Consider two lines with direction cosines \(l, m, n\) and \(a, b, c\). It is stated that:

   The lines are parallel if: \(\frac{l}{a} = \frac{m}{b} = \frac{n}{c}\)

   This condition describes proportionality between the direction ratios, hence making lines parallel. This statement is incorrect in the context provided above where it was provided as a condition for perpendicularity.

2. The lines are perpendicular if: \(la + mb + nc = 0\)

   This condition is derived from the dot product of two vectors being zero, thus indicating orthogonality. This statement is essentially about perpendicularity, contradicting the initial proposition.

3. As per the options, the correct answer is: "the direction ratios of the bisectors of the angles between l±a, m±b, n±c".

   Let's examine what this means:

   The direction cosines of the bisectors of the angles between two lines can be expressed as either the sum or difference of their direction cosines, which is what's encapsulated by the expression \(l \pm a\), \(m \pm b\), and \(n \pm c\).

   Thus, this statement correctly refers to the direction cosines of the angle bisectors, encompassing conditions for both sum and difference.

From the analysis above, it is clear that the expression regarding the direction ratios of the bisectors of angles between \(l\pm a\), \(m\pm b\), \(n\pm c\) indeed correctly describes how these bisectors would appear in terms of direction cosines. Hence, it's the correct choice.