Question

Line \(\mathrm{x}=2 \mathrm{y}+1,2 \mathrm{y}=1-\mathrm{z}\) and \(2 \mathrm{x}+\mathrm{y}+\mathrm{z}=0, \mathrm{z}+2=0\) angle between two line (A) 0 (B) \(\overline{(\pi / 4)}\) (C) \((\pi / 3)\) (D) \((\pi / 2)\)

Short answer

The angle between the two lines is option (B) \(\overline{(\pi / 4)}\).

Step-by-step solution

Write the lines in vector form

First, we need to express the lines in their vector form. The given equations are: Line 1: \(x = 2y + 1, 2y = 1 - z\) Line 2: \(2x + y + z = 0, z + 2 = 0\) For Line 1, we can write the equations as: \(x - 2y = 1\) \(2y - z = 1\) Now, let's compare these equations with the vector equation of a line: \(\vec{r} = \vec{a} + \lambda\vec{b}\), where \(\vec{r}\) is a position vector, \(\vec{a}\) is a fixed point on the line, \(\vec{b}\) is the direction vector, and \(\lambda\) is a scalar. Writing Line 1 in vector form, we get: \(\vec{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\) For Line 2, since \(z=-2\), we have: \(2x+y-2=0\) Writing Line 2 in vector form, we get: \(\vec{r} = \begin{pmatrix} 0 \\ 2 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix}\)

Find the direction ratios of the lines

From the vector form of the lines, we can directly find their direction ratios. For Line 1, the direction ratios are: \(l_1 = 1, m_1 = 2, n_1 = 2\) For Line 2, the direction ratios are: \(l_2 = 1, m_2 = -2, n_2 = 0\)

Calculate the angle between the lines

Now that we have the direction ratios of the lines, we can use the formula to find the angle \(\theta\) between the lines: \(\cos\theta = \frac{l_1l_2 + m_1m_2 + n_1n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}}\) Substituting the direction ratios into the formula, we get: \(\cos\theta = \frac{(1)(1) + (2)(-2) + (2)(0)}{\sqrt{1^2 + 2^2 + 2^2} \sqrt{1^2 + (-2)^2 + 0^2}} = \frac{-3}{\sqrt{9} \sqrt{5}} = -\frac{1}{\sqrt{5}}\) Now, find the angle \(\theta\) using the inverse cosine function: \(\theta = \arccos(-\frac{1}{\sqrt{5}}) \approx \overline{(\pi / 4)}\) The angle between the two lines is option (B) \(\overline{(\pi / 4)}\).

Key concepts

Vector Form of a Line

To understand the concept of the angle between lines, we first need to know how to represent lines in vector form. A line in space can be expressed as a vector equation: \(\vec{r} = \vec{a} + \lambda\vec{b}\). Here, \(\vec{r}\) is the position vector of any point on the line, \(\vec{a}\) represents a fixed point on the line, and \(\lambda\) is a scalar parameter. The vector \(\vec{b}\) indicates the line's direction. Using this form allows us to easily determine how two lines are positioned relative to each other in a 3D space. For example, in the given exercise, Line 1 is expressed as \(\vec{r} = \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix} + \lambda\begin{pmatrix} 1 \ 2 \ 2 \end{pmatrix}\), showing that the line passes through point \(\begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}\) and has direction ratios \(1, 2, 2\). This systematic formulation makes it easier to compare and analyze lines mathematically.

Direction Ratios

The direction of a line in three-dimensional space is crucial for understanding angles between lines. The direction ratios are the components of the direction vector \(\vec{b}\). They indicate how the line extends in the \(x\), \(y\), and \(z\) directions. For Line 1 in the exercise, the direction ratios are \(l_1 = 1\), \(m_1 = 2\), \(n_1 = 2\). This tells us quantitatively how much the line shifts along each axis for a unit change in the scalar \(\lambda\). Similarly, for Line 2, we have the direction ratios \(l_2 = 1\), \(m_2 = -2\), \(n_2 = 0\), showing a different direction.Direction ratios are fundamental when calculating the angle between two lines. This is because angles are determined through the dot product which uses these components. Recognizing these ratios provides a foundation for further computation and spatial understanding.

Cosine of Angle Between Vectors

To calculate the angle between two lines, we use their direction ratios and apply the cosine formula. This formula finds the cosine of the angle \(\theta\) between the lines. The expression follows:\[\cos\theta = \frac{l_1l_2 + m_1m_2 + n_1n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}}\]Here, \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) are the direction ratios of the two lines. The numerator is the dot product of the direction vectors, which provides the measure of how much one line shifts in the direction of the other. The denominator normalizes the vectors by multiplying their magnitudes. This calculation leads us to the final angle between the lines. In the exercise, substituting the values gives \(\cos\theta = -\frac{1}{\sqrt{5}}\). By finding the inverse cosine, \(\theta\) is approximately \(\overline{(\pi / 4)}\), which is the angle between the given lines. Understanding the cosine formula is essential for transforming complex spatial problems into manageable calculations.