#### Question

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

#### Solution

\[ \text{ The direction ratios of the first line are 1, - 2, 1 and the direction ratios of the second line are 4, 3, 2 } . \]

\[ \text{ Let } \theta \text{ be the angle between these two lines } . \]

\[\text{ Now }, \]

\[\cos \theta = \left| \frac{1\left( 4 \right) + \left( - 2 \right)\left( 3 \right) + 1\left( 2 \right)}{\sqrt{\left( 1 \right)^2 + \left( - 2 \right)^2 + \left( 1 \right)^2} \sqrt{\left( 4 \right)^2 + \left( 3 \right)^2 + \left( 2 \right)^2}} \right|\]

\[ = \left| \frac{4 - 6 + 2}{\sqrt{1 + 4 + 1}\sqrt{16 + 9 + 4}} \right|\]

\[ = \frac{0}{\sqrt{6}\sqrt{29}}\]

\[ = 0 \]

\[ \Rightarrow \theta = \frac{\pi}{2}\]

\[\text { Hence, the required angle is } \frac{\pi}{2} .\]