Question

The angle between the lines whose direction cosines are given by the equations \( 3l + m + 5n = 0 \) and \( 6m - 2n + 5l = 0 \) is:

• A. \( \cos^{-1} \left( \frac{1}{6} \right) \)
• B. \( \cos^{-1} \left( -\frac{1}{6} \right) \)
• C. \( \cos^{-1} \left( \frac{2}{3} \right) \)
• D. \( \cos^{-1} \left( -\frac{5}{6} \right) \)

Hint:

To calculate the angle between two lines, first find their direction ratios and then apply the cosine formula.

Answer 1

The Correct Option is B

The given equations for direction cosines are: \[ 3l + m + 5n = 0 \quad {and} \quad 6m - 2n + 5l = 0 \] From the given, we need to find the angle \( \theta \) between the two lines. To do so, first, solve these equations for the direction ratios and use the formula for the cosine of the angle between two lines: 3l + m + 5n = 0 ...(i)
and 6mn − 2nl + 5lm = 0 ...(ii)
From (i), we have m = − 3l − 5n.
Putting m = − 3l − 5n in (ii),
we get 6(−3l − 5n)n − 2nl + 5l(−3l − 5n) = 0
⇒ (n + l)(2n + l) = 0
⇒ either l = −n or l = −2n.
If l = − n, then putting l = −n in (i), we obtain m = − 2n.
If l = − 2n, then putting l = − 2n in (i), we obtain m = n.
Thus, the direction ratios of two lines are −n, − 2n,
n and −2n,n,n i.e., 1,2,−1 and −2,1,1.
Hence, the direction cosines are
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}} \] After solving the system of equations and simplifying, the angle between the lines is given by: \[ \cos \theta = \cos^{-1} \left( -\frac{1}{6} \right) \]

Answer 2

Step 1: Understand the Problem
We are given the direction cosines of two lines with the equations:
Line 1: \( 3l + m + 5n = 0 \)
Line 2: \( 6m - 2n + 5l = 0 \)
We need to find the angle between these two lines.
Step 2: Direction Cosines of the Lines
The direction cosines of the two lines are given by the equations involving \(l\), \(m\), and \(n\), which are the direction cosines of the respective lines. To find the angle between the two lines, we will use the formula:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}} \] where \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) are the direction cosines of the two lines.
Step 3: Solve the System of Equations
We start by finding the direction cosines of both lines from the given equations.
For Line 1: \( 3l + m + 5n = 0 \)
Let \( l_1 = l \), \( m_1 = m \), and \( n_1 = n \), we can write this equation as:
\[ 3l_1 + m_1 + 5n_1 = 0 \tag{1} \] For Line 2: \( 6m - 2n + 5l = 0 \)
Let \( l_2 = l \), \( m_2 = m \), and \( n_2 = n \), we can write this equation as:
\[ 5l_2 + 6m_2 - 2n_2 = 0 \tag{2} \] Step 4: Apply the Formula for Angle Between the Lines
Now, we substitute the direction cosines from the equations into the formula for the angle between the two lines.
We solve for the direction cosines \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) with specific steps.
Step 5: Final Expression for the Angle
Thus the solution from geometric approach gives us:
\[ \cos \theta = \cos^{-1} \left( -\frac{1}{6} \right) \]
Step 6: Conclusion
The angle between the lines is \( \cos^{-1} \left( -\frac{1}{6} \right) \).