Question

Find the angle between $\dfrac{x}{{ - 1}} = \dfrac{{y - 2}}{2} = \dfrac{{z - 3}}{6}$ and $\dfrac{{x + 1}}{6} = \dfrac{{y + 2}}{2} = \dfrac{{z - 1}}{3}$.

Answer 1

Here, we need to find the angle between the two given lines. We will compare the given equations with the standard form in cartesian form and find the direction ratios of the two lines. Then, we will substitute the direction ratios obtained in the formula for the angle between two lines and simplify to get the answer.
Formula Used: The angle $\theta$ between two lines $\dfrac{{x - {x_1}}}{{{a_1}}} = \dfrac{{y - {y_1}}}{{{b_1}}} = \dfrac{{z - {z_1}}}{{{c_1}}}$ and $\dfrac{{x - {x_2}}}{{{a_2}}} = \dfrac{{y - {y_2}}}{{{b_2}}} = \dfrac{{z - {z_2}}}{{{c_2}}}$ is given by $\cos \theta = \left| {\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\left( {\sqrt {a_1^2 + b_1^2 + c_1^2} } \right)\left( {\sqrt {a_2^2 + b_2^2 + c_2^2} } \right)}}} \right|$, ${a_1}$, ${b_1}$, and ${c_1}$ are the direction ratios of the first line and ${a_2}$, ${b_2}$, and ${c_2}$ are the direction ratios of the second line.

Complete step by step solution:
The standard form of an equation of a line in cartesian form where the line passes through the point $\left( {{x_1},{y_1},{z_1}} \right)$ is $\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c}$, where $a$, $b$, and $c$ are the direction ratios of the line.
We will compare the given equations with the standard form of an equation of a line in cartesian form to find the direction ratios of the two lines.
Let ${a_1}$, ${b_1}$, and ${c_1}$ be the direction ratios of the first line and ${a_2}$, ${b_2}$, and ${c_2}$ be the direction ratios of the second line.
Comparing the line $\dfrac{x}{{ - 1}} = \dfrac{{y - 2}}{2} = \dfrac{{z - 3}}{6}$ with the standard form $\dfrac{{x - {x_1}}}{{{a_1}}} = \dfrac{{y - {y_1}}}{{{b_1}}} = \dfrac{{z - {z_1}}}{{{c_1}}}$, we get
${a_1} = - 1$, ${b_1} = 2$, and ${c_1} = 6$
Comparing the line $\dfrac{{x + 1}}{6} = \dfrac{{y + 2}}{2} = \dfrac{{z - 1}}{3}$ with the standard form $\dfrac{{x - {x_2}}}{{{a_2}}} = \dfrac{{y - {y_2}}}{{{b_2}}} = \dfrac{{z - {z_2}}}{{{c_2}}}$, we get
${a_2} = 6$, ${b_2} = 2$, and ${c_2} = 3$
Now, we will substitute the values of the direction ratios in the formula for angle between two lines.
The angle $\theta$ between two lines $\dfrac{{x - {x_1}}}{{{a_1}}} = \dfrac{{y - {y_1}}}{{{b_1}}} = \dfrac{{z - {z_1}}}{{{c_1}}}$ and $\dfrac{{x - {x_2}}}{{{a_2}}} = \dfrac{{y - {y_2}}}{{{b_2}}} = \dfrac{{z - {z_2}}}{{{c_2}}}$ is given by $\cos \theta = \left| {\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\left( {\sqrt {a_1^2 + b_1^2 + c_1^2} } \right)\left( {\sqrt {a_2^2 + b_2^2 + c_2^2} } \right)}}} \right|$, where ${a_1}$, ${b_1}$, and ${c_1}$ are the direction ratios of the first line and ${a_2}$, ${b_2}$, and ${c_2}$ are the direction ratios of the second line.
Substituting ${a_1} = - 1$, ${b_1} = 2$, ${c_1} = 6$, ${a_2} = 6$, ${b_2} = 2$, and ${c_2} = 3$, we get
$\cos \theta = \left| {\dfrac{{\left( { - 1 \cdot 6} \right) + \left( {2 \cdot 2} \right) + \left( {6 \cdot 3} \right)}}{{\left( {\sqrt {{{\left( { - 1} \right)}^2} + {2^2} + {6^2}} } \right)\left( {\sqrt {{6^2} + {2^2} + {3^2}} } \right)}}} \right|$
Multiplying the terms in the expression, we get
$\Rightarrow \cos \theta = \left| {\dfrac{{ - 6 + 4 + 18}}{{\left( {\sqrt {1 + 4 + 36} } \right)\left( {\sqrt {36 + 4 + 9} } \right)}}} \right|$
Adding the terms in the expression, we get
$\Rightarrow \cos \theta = \left| {\dfrac{{16}}{{\left( {\sqrt {41} } \right)\left( {\sqrt {49} } \right)}}} \right|$
Simplifying and rewriting the equation, we get
$\begin{array}{l}\\\ \Rightarrow \cos \theta = \left| {\dfrac{{16}}{{7\sqrt {41} }}} \right|\\\ \Rightarrow \cos \theta = \dfrac{{16}}{{7\sqrt {41} }}\\\ \Rightarrow \theta = {\cos ^{ - 1}}\left( {\dfrac{{16}}{{7\sqrt {41} }}} \right)\end{array}$

Therefore, the angle between the given lines is ${\cos ^{ - 1}}\left( {\dfrac{{16}}{{7\sqrt {41} }}} \right)$.

Note:
Here, we have used the standard form of the equation of a line in cartesian form. The standard equation of a line in cartesian form where the line passes through the point $\left( {{x_1},{y_1},{z_1}} \right)$ is $\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c}$, where $a$, $b$, and $c$ are the direction ratios of the line.