Kerala KEAM 5 June 2024 Solved Paper

We are given two straight lines in parametric form: $$\overrightarrow{r_1} = (4i - k) + t(2i + j - 2k), \quad t \in \mathbb{R},$$and $$\overrightarrow{r_2} = (i - j + 2k) + s(2i - 2j + k), \quad s \in \mathbb{R}.$$Step 1: Find the direction ratios of the two lines.For $\overrightarrow{r_1}$, the direction ratios are the coefficients of $t$, which are: $$\overrightarrow{a_1} = 2i + j - 2k.$$For $\overrightarrow{r_2}$, the direction ratios are the coefficients of $s$, which are: $$\overrightarrow{a_2} = 2i - 2j + k.$$Step 2: Use the formula for the angle between two vectors: $$\cos\theta = \frac{\overrightarrow{a_1} \cdot \overrightarrow{a_2}}{ | \overrightarrow{a_1} | \overrightarrow{a_2} | }.$$The dot product $\overrightarrow{a_1} \cdot \overrightarrow{a_2}$ is: $$\overrightarrow{a_1} \cdot \overrightarrow{a_2} = (2)(2) + (1)(-2) + (-2)(1) = 4 - 2 - 2 = 0.$$Since the dot product is 0, the vectors are perpendicular, which means the angle $\theta$ between the two vectors is: $$\theta = \frac{\pi}{2}.$$Thus, the angle between the two lines is $\frac{\pi}{2}$, and the correct answer is option (E).Quick Tip: When two vectors are perpendicular, their dot product is zero, and the angle between them is $\frac{\pi}{2}$ radians (90 degrees). This can be used to find the angle between two lines.