#### Question

If the coordinates of the points *A*, *B*, *C*, *D* are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between *AB* and *CD*.

#### Solution

\[\text { The given points are } A \left( 1, 2, 3 \right), B\left( 4, 5, 7 \right), C\left( - 4, 3, - 6 \right) \text{ and } D \left( 2, 9, 2 \right) . \]

\[\text { We know that the direction ratios of the line joining the points } \left( x_1 , y_1 , z_1 \right) \text { and } \left( x_2 , y_2 , z_2 \right) \text { are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]

\[\text { The direction ratios of AB are } \left( 4 - 1 \right), \left( 5 - 2 \right), \left( 7 - 3 \right), \text { i . e } . 3, 3, 4 . \]

\[\text { The direction ratios of CD are } \left[ 2 - \left( - 4 \right) \right], \left( 9 - 3 \right), \left[ 2 - \left( - 6 \right) \right], \text { i . e }. 6, 6, 8 . \]

\[\text { Let } \theta \text { be the angle between AB and CD } . \]

\[\text { We have } \]

\[ a_1 = 3, b_1 = 3, c_1 = 4 \]

\[ a_2 = 6, b_2 = 6, c_2 = 8\]

\[ \therefore \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\sqrt{{a_2}^2 + {b_2}^2 + {c_2}^2}} = \frac{18 + 18 + 32}{\sqrt{9 + 9 + 16}\sqrt{36 + 36 + 64}} = \frac{68}{68} = 1\]

\[ \Rightarrow \theta = 0° \]

\[\text { Thus, the angle between AB and CD measures } 0° . \]