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

Two lines having direction ratios $a_1,b_1,c_1$ and $a_2,b_2,c_2$ are parallel if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Also the line passing through $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ has the direction ratio $(x_2-x_1),(y_2-y_1),(z_2-z_1)$

$A(4,7,8)$

$B(2,3,4)$

D.r's $=2-4,3-7,4-8$

$=-2,-4,-4$

$\therefore$ $a_1=-2;b_1=-4,c_1=-4$

$C(-1,-2,1)$

$D(1,2,5)$

Dr's $=1-(-1),2-(-2),5-1$

$=2,4,4$

$\therefore$ $a_2=2,b_2=4,c_2=4$

Now $\frac{a_1}{a_2}=\frac{-2}{2}=-1$; $\frac{b_1}{b_2}=\frac{-4}{4}$; $\frac{c_1}{c_2}=\frac{-4}{4}=-1$

$\therefore$ Given lines are parallel