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

Toolbox:
• For two lines to be parallel, their direction ration should be proportional. (ie) $\large \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Let the points be $A(4,7,8), B(2,3,4),C(-1,-2,1)$ and $D(1,2,5)$
The direction rations of the line joining $A$ and $B$ is $(2-4),(3,-7),(4-8)$
On simplifying we get,
$(-2,-4,-4)$ (let this be $(a_1,b_1,c_1))$
The direction ratios of the line joining C and D is
$(1-(-1)),(2-(-2)),(5,-1)$
On simplifying we get,
$(2,4,4)$ (let this be $(a_2,b_2,c_2))$
For the line to be parallel,
$\large \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Now substituting for $a_1,b_1,c_1$ and $a_2,b_2,c_2$ we get
$\large\frac{-2}{2}=\frac{-4}{4}=\frac{-4}{4}=1$
Since the ratios are equal, the lines are parallel.