Question

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

Answer 1

Let the given points be A(4, 7, 8), B(2, 3,4), C(- 1, - 2,1) and D(1, 2, 5).
Direction ratios of $AB(a_1,b_1,c_1)$ are (2 - 4), (3 - 7), (4 - 8) $\Rightarrow$ (- 2, - 4, - 4)
$\therefore a_1=-2,b_1=-4,c_1=-4$
Direction ratios of $CD(a_2,b_2,c_2)$ are (1 + 1), (2 + 2), (5 -1) $\Rightarrow$ (2, 4,4)
$a_2=2,b_2=4,c_2=4$
AB is parallel to CD if
$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
We have $\frac{a_1}{a_2}=\frac{-2}{2}=-1,\frac{b_1}{b_2}=\frac{-4}{4}=-1and\frac{c_1}{c_2}=\frac{-4}{4}=-1$
$\therefore \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ Hence, AB II CD.