# 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 $CD$.

$\begin{array}{1 1} (A) \;0^{\large\circ}\; or \;180^{\large\circ} \\ (B) \;90^{\large\circ}\;or \;270^{\large\circ} \\ (C)\; 60^{\large\circ}\; or \;120^{\large\circ} \\ (D)\; 30^{\large\circ}\; or \;150^{\large\circ} \end{array}$

Toolbox:
• If two lines are parallel,then $\large\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
• Where $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$ are the direction ratios of the two lines.
Step 1:
Given the coordinates of $A,B,C$ and $D$ are $(1,2,3),(4,5,7),(-4,3,-6)$ and $(2,9,2)$ respectively.
We know the direction ratios of $AB$ are $(x_2-x_1),(y_2-y_1),(z_2-z_1)$
(i.e) (4-1),(5-2),(7-3)
$\Rightarrow (3,3,4)$
Step 2:
The direction ratios of $CD$ are $(x_4-x_3),(y_4-y_3),(z_4-z_3)$
(i.e) $(2-(-4)),(9-3),(2-(-6))$
$\Rightarrow (6,6,8)$
Therefore $\large\frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2}$
$\large\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$
$\large\frac{c_1}{c_2}=\frac{4}{8}=\frac{1}{2}$
Step 3:
Hence we infer that $AB$ is parallel to $CD$
Therefore the angle between $AB$ and $CD$ is $0^{\large\circ}$ or $180^{\large\circ}$