Question

Find the angle between the pairs of lines $\dfrac{{x + 3}}{3} = \dfrac{{y - 1}}{5} = \dfrac{{z + 3}}{4}$ and $\dfrac{{x + 1}}{1} = \dfrac{{y - 4}}{1} = \dfrac{{z - 5}}{2}$

Answer 1

Hint : Here we are given the cartesian equation of the line. The cartesian equation is used to represent lines/curves or any other surface on a graph paper. Equation of a line which has a passing point $({x_1},{y_1},{z_1})$ and is parallel to a line whose direction cosines are a, b and c is given as
$\dfrac{{x - {x_1}}}{a} = \dfrac{{y - {y_1}}}{b} = \dfrac{{z - {z_1}}}{c}$ . So from the given equation of the line, we can find out the direction cosines and thus the angle between the lines.

Complete step-by-step answer :
Equation of the first line is $\dfrac{{x + 3}}{3} = \dfrac{{y - 1}}{5} = \dfrac{{z + 3}}{4}$ , so we have $\overrightarrow {{b_1}} = 3\hat i + 5\hat j + 4\hat k$
Equation of the second line is
$\dfrac{{x + 1}}{1} = \dfrac{{y - 4}}{1} = \dfrac{{z - 5}}{2}$ , so we have
$\overrightarrow {{b_2}} = \hat i + \hat j + 2\hat k$
On rearranging the formula of dot product of the above two vectors, we get
$\cos \theta = \dfrac{{{{\vec b}_1}{{\vec b}_2}}}{{\left| {{{\vec b}_1}} \right|\left| {{{\vec b}_2}} \right|}}$
Where $\left| {{{\vec b}_1}} \right|$ is the magnitude of ${\vec b_1}$ and $\left| {{{\vec b}_2}} \right|$ is the magnitude of ${\vec b_2}$ .
The magnitude of a vector $\vec a = x\hat i + y\hat j + z\hat k$ is given as $\left| a \right| = \sqrt {{x^2} + {y^2} + {z^2}}$
Using this we get,
$\Rightarrow \cos \theta = \dfrac{{(3\hat i + 4\hat j + 5\hat k).(\hat i + \hat j + 2\hat k)}}{{\sqrt {{3^2} + {5^2} + {4^2}} \times \sqrt {{1^2} + {1^2} + {2^2}} }} = \dfrac{{3 + 4 + 10}}{{\sqrt {50} \times \sqrt 6 }} \\\ \Rightarrow \cos \theta = \dfrac{{16}}{{10\sqrt 3 }} = \dfrac{8}{{5\sqrt 3 }} \\\ or\,\theta = {\cos ^{ - 1}}(\dfrac{8}{{5\sqrt 3 }}) \;$
Thus, the angle between the given two lines is ${\cos ^{ - 1}}(\dfrac{8}{{5\sqrt 3 }})$ .
So, the correct answer is “ ${\cos ^{ - 1}}(\dfrac{8}{{5\sqrt 3 }})$ ”.

Note : In parameter equations, the dependent variables that are x and y are linked to an independent variable called parameter that is often denoted by t. Unlike parametric equations, Cartesian equations tell the relation between only x and y coordinates.
Dot product is the product of two vectors to convert the length sequence of numbers into a single number. It is the product of the magnitude of the two vectors and the cosine of the angle between them. We have used the formula of dot product in this question to find out the angle between the given lines.