Problem 8 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The angle between the lines whose direction cosines satisfy the equations \(l+m+n=0\) and \(l^{2}+m^{2}+n^{2}\) is [2014] (a) \(\frac{\pi}{6}\) (b) \(\frac{\pi}{2}\) (c) \(\frac{\pi}{3}\) (d) \(\frac{\pi}{4}\)

Step-by-Step Solution

Answer

Angle is \(\frac{\pi}{3}\) or 60 degrees.

Step 1: Understand the given conditions

The direction cosines of a line are given by \(l, m, n\) and are subject to the equations \(l+m+n=0\) and \(l^2 + m^2 + n^2 = 1\). The problem is to find the angle between two different lines whose direction cosines satisfy these equations.

Step 2: Use properties of direction cosines

For two lines with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), the cosine of the angle \(\theta\) between them is given by the dot product formula: \(l_1 l_2 + m_1 m_2 + n_1 n_2 = \cos \theta\).

Step 3: Substitute known conditions

Since \(l + m + n = 0\), rearrange to express one in terms of the others, say \(n = -(l+m)\). Substitute this into the second equation, \(l^2 + m^2 + (-(l+m))^2 = 1\), and expand it: \(l^2 + m^2 + (l^2 + 2lm + m^2) = 1\).

Step 4: Simplify and calculate possible values

The expanded equation becomes \(2l^2 + 2m^2 + 2lm = 1\). Dividing the entire equation by 2 gives \(l^2 + m^2 + lm = \frac{1}{2}\). Given the symmetric property and solutions for direction cosines, try possible symmetric values.

Step 5: Calculate cosine using symmetry

We consider symmetrical lines like \((l, m, -(l+m))\) and \( (m, l, -(m+l))\). Substituting \(l, m\) as \(1/2, -1/2\): \( l^2 \approx m^2 \). The angle between these is \(\cos \theta = \frac{-1}{2}\), fulfilling the symmetry.

Step 6: Determine the angle

With \(\cos \theta = -\frac{1}{2}\), the angle \(\theta\) is \(\cos^{-1}(-\frac{1}{2})\) which corresponds to \(\frac{3\pi}{2}\). Therefore, the angle between such lines is complementary to \(\pi/3\).

Key Concepts

Direction Cosines

When we talk about direction cosines, we are referring to the cosines of the angles that a line makes with the positive directions of the coordinate axes. These are typically denoted as \(l, m, n\). The direction cosines have a crucial relationship given by the equation \(l^2 + m^2 + n^2 = 1\). This equation ensures that the vector defined by these cosines is a unit vector.

• \(l\) is the cosine of the angle between the line and the x-axis.
• \(m\) is the cosine of the angle with the y-axis.
• \(n\) is the cosine related to the z-axis.

These variables help specify the orientation of a line in 3D space. Whenever given an additional equation like \(l + m + n = 0\), it further constrains the relationship between these cosines, which can then be used to analyze or find relationships like angles between lines.

Angle Between Lines

The angle between two lines in space is a common requirement in geometry and physics. The relationship of direction cosines aids in this calculation. For two lines with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), we use their dot product to find the angle \(\theta\) between them.
The cosine of the angle between any two lines can be derived using:\[\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2\]
In simple terms:

• This formula is based on projecting one vector onto another.
• If \(\cos \theta\) is positive, the lines form an acute angle, and if negative, they form an obtuse angle.
• Special cases include perpendicular lines, where \(\cos \theta = 0\), indicating the dot product is zero.

To solve this specific problem, the condition \(l + m + n = 0\) was used to further investigate the symmetry and find symmetric lines that satisfy this condition, leading to an exact angle.

Dot Product Formula

The dot product, also known sometimes as the scalar product, is an algebraic operation that involves two equal-length sequences of numbers. It is important in physics and engineering for determining the angle between two vectors in space.

• The dot product of two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\) is \(a_1 b_1 + a_2 b_2 + a_3 b_3\).
• This operation results in a scalar and is foundational when determining angles using direction cosines.

The formula for the dot product ties directly into finding the angle \(\theta\) between two lines with:\[\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\]

• If \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are unit vectors, the formula simplifies as seen in our earlier discussion of the angle between lines.

In this scenario, once the direction cosines have been fully understood and substituted, the dot product formula allows for precise calculation of \(\theta\). The problem ultimately used the established equations and symmetry to calculate the correct angle alternatives like \(\frac{\pi}{3}\), highlighting the role of the dot product in geometrical interpretation.