Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Home » Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Solution:

Given that
The points $(1,-1,2),(3,4,-2)$ and $(0,3,2),(3,5,6)$ .
Let’s consider $A B$ be the line joining the points, $(1,-1,2)$ and $(3,4,-2)$ , and $C D$ be the line through the points $(0,$ , $3,2)$ and $(3,5,6)$ .
So now,
Direction ratios, $a_{1}, b_{1}, c_{1}$ of $A B$ are
$(3-1),(4-(-1)),(-2-2)=2,5,-4$
In the similar way,
The direction ratios, $a_{2}, b_{2}, c_{2}$ of $C D$ are
$(3-0),(5-3),(6-2)=3,2,4$
Therefore, $A B$ and $C D$ will be perpendicular to each other, if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$
$\begin{array}{l} a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=2(3)+5(2)+4(-4) \\ =6+10-16 \\ =0 \end{array}$
As a result, $\mathrm{AB}$ and $\mathrm{CD}$ are perpendicular to each other.