Question

If the coordinates of the points $A,B,C,D$ be $(1,2,3),(4,5,7),(-4,3,-6)$ and $(2,9,2)$ respectively, then find the angle between the lines $AB$ and $CD$

Answer 1

The coordinates of $A,B,C$ and $D$ are $(1,2,3),(4,5,7),(-4,3,-6)$ and $(2,9,2)$ respectively.
The direction ratios of $AB$ are $(4-1)=3,(5-2)=3$ and $(7-3)=4$
The direction ratios of $CD$ are $(2-(-4\left)\right)=6,(9-3)=6$ and $(2-(-6\left)\right)=8$
It can be seen that $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{1}{2}$
Therefore, $AB$ is parallel to $CD$.
Thus, the angle between $AB$ and $CD$ is either $0^o$ or ${180}^o$.