Show that the line through points (4, 7, 8), (2, 3, 4) is paralle… - Vidyadip

Question

Show that the line through points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (-1, -2, 1), (1, 2, 5).

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Answer

We know that direction ratios of the line joining the points A (4, 7, 8) and B (2, 3, 4) are ${x_2} - {x_1}, {y_2} - {y_1},{z_2} - {z_1}$
$ \Rightarrow 2 - 4,3 - 7,4 - 8$
$ \Rightarrow - 2, - 4, - 4 = {a_1},{b_1},{c_1}$ (say)
Again direction ratios of the line joining the points $C\left( { - 1, - 2,1} \right)$ and D (1, 2, 5) are $x_2-x_1,y_2-y_1,z_2-z_1$
$\Rightarrow 1 - \left( { - 1} \right),2 - \left( { - 2} \right),5 - 1$
$ \Rightarrow 2,4,4 = {a_2},{b_2},{c_2}$(say)
For the lines AB and CD, $\frac{{{a_1}}}{{{a_2}}} = \frac{{ - 2}}{2},\frac{{{b_1}}}{{{b_2}}} = \frac{{ - 4}}{4},\frac{{{c_1}}}{{{c_2}}} = \frac{{ - 4}}{4} = - 1$
Since, $\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$
Therefore, line AB is parallel to line CD.

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