Question

If the coordinates of the point A, B, C, D be $(1,2,3),(4,5,7),( - 4,3, - 6)$and $\left( {2,9,2} \right)$respectively. Then find the angle between the line AB and CD

Hint: Write the direction ratios of both the lines.

The coordinates of A, B, C, D are given to us as $(1,2,3),(4,5,7),( - 4,3, - 6)$and $\left( {2,9,2} \right)$respectively.
If a line passes through two points that is $\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right)$then the direction ratio of that line can be written as ${x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}$
Using this concept we can write the direction ratios of line AB as $\left( {4 - 1),(5 - 2),(7 - 3} \right)$which implies
$\Rightarrow \left( {{a_1},{b_1},{c_1}} \right) = 3,3,4$
Now the direction ratios of line CD is $\left( {2 - ( - 4)),(9 - 3),(2 - ( - 6)} \right)$
$\Rightarrow \left( {{a_2},{b_2},{c_2}} \right) = 6,6,8$
Now if two lines are parallel then their direction ratios satisfy $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} = \lambda$then lines are parallel
So using the above concept we have $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$as $\dfrac{3}{6} = \dfrac{3}{6} = \dfrac{4}{8}$which is

$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{1}{2}$
Hence line AB is parallel to CD
Thus the angle between AB and CD is either ${0^\circ }{\text{ or 18}}{{\text{0}}^\circ }$.

Note- These category of questions is always approached using the concept of direction ratios as being stated above, now the point to remember is if the direction ratios are following the equation that $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} = \lambda$ then both lines are said to be parallel.