Question

If the co-ordinates of the points P, Q, R, S be \[\left( {1,2,3} \right),\left( {4,5,7} \right),\left( { - 4,3, - 6} \right)\] and $\left( {2,0,2} \right)$ then
A. PQ $\parallel $ RS
B. PQ $ \bot $ RS
C. PQ $ = $ RS
D. None of these

Answer 1

Hint: In order to solve this type of question we will use the formula of finding the angle between the lines. First we will find the direction ratios of both the lines separately then we will substitute the values of the direction ratios obtained in the formula $\cos \theta = \left| {\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} \sqrt {{a_2}^2 + {b_2}^2 + {c_2}^2} }}} \right|$ for finding the angle between the lines to get the correct answer.

Formula used:
Angle between a pair of lines having direction ratios ${a_1},{b_1},{c_1}$ and ${a_2},{b_2},{c_2}$ is given by,
$\cos \theta = \left| {\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} \sqrt {{a_2}^2 + {b_2}^2 + {c_2}^2} }}} \right|$
Direction ratio of line passing through points $A\left( {{x_1},{y_1},{z_1}} \right)$ and $B\left( {{x_2},{y_2},{z_2}} \right)$ is calculated by $\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$.
Complete step by step solution:
We will first calculate the direction ratios of both the lines PQ and RS separately. We know that direction ratios of the line passing through points $A\left( {{x_1},{y_1},{z_1}} \right)$ and $B\left( {{x_2},{y_2},{z_2}} \right)$ is calculated by $\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$.
Direction ratio of line PQ passing through points $P\left( {1,2,3} \right)$ and $Q\left( {4,5,7} \right)$.
$\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$
Substituting the values,
$\left( {4 - 1} \right),\left( {5 - 2} \right),\left( {7 - 3} \right) = 3,3,4$
$\therefore {a_1} = 3,\;{b_1} = 3,\;{c_1} = 4$ ………………..equation $\left( 1 \right)$
Direction ratio of line RS passing through points $R\left( { - 4,3, - 6} \right)$ and $S\left( {2,0,2} \right)$.
$\left( {{x_2} - {x_1}} \right),\left( {{y_2} - {y_1}} \right),\left( {{z_2} - {z_1}} \right)$
Substituting the values,
$\left( {2 - \left( { - 4} \right)} \right),\left( {0 - 3} \right),\left( {2 - \left( { - 6} \right)} \right) = 6, - 3,8$
$\therefore {a_2} = 6,\;{b_2} = - 3,\;{c_2} = 8$ ………………..equation $\left( 2 \right)$
Substituting equations $\left( 1 \right)$ and $\left( 2 \right)$ in angle formula.
$\cos \theta = \left| {\dfrac{{\left( {3 \times 6} \right) + \left( {3 \times \left( { - 3} \right)} \right) + \left( {4 \times 8} \right)}}{{\sqrt {{3^2} + {3^2} + {4^2}} \sqrt {{6^2} + {{\left( { - 3} \right)}^2} + {8^2}} }}} \right|$
Solving it,
$\cos \theta = \left| {\dfrac{{18 - 9 + 32}}{{\sqrt {34} \sqrt {109} }}} \right|$
$\cos \theta = \left| {\dfrac{{41}}{{\sqrt {3706} }}} \right|$
Finding $\theta ,$
$\theta = {\cos ^{ - 1}}\left| {\dfrac{{41}}{{\sqrt {3706} }}} \right|$
$\theta = 0.832$
Thus, we conclude that neither PQ $\parallel $ RS nor PQ $ \bot $ RS. Also, PQ $ \ne $ RS.
$\therefore $ The correct option is (D).

Note: The direction ratios are very helpful in finding the relationship between two lines or vectors. The direction ratios can be used to find the direction cosines of a line or the angle between the two lines. The direction ratios are also useful in finding the dot product between the two vectors.