Question

The angle between the two straight lines \[ \overrightarrow{r_1} = (4i - k) + t(2i + j - 2k), \quad t \in \mathbb{R}, \quad {and} \quad \overrightarrow{r_2} = (i - j + 2k) + s(2i - 2j + k), \quad s \in \mathbb{R} \] is:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{6} \)
• D. \( 0 \)
• E. \( \frac{\pi}{2} \)

Hint:

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.

Answer 1

Answer: (E)

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).