Question

The angle between the lines $\vec r = (2\hat i + \hat j - 3\hat k) + \lambda(\hat i - \hat j + \hat k)$ and $\dfrac{x-1}{1} = \dfrac{y+2}{3} = \dfrac{z-3}{2}$ is

• A. $\dfrac{\pi}{6}$
• B. $\dfrac{\pi}{3}$
• C. $\dfrac{\pi}{4}$
• D. $\dfrac{\pi}{2}$

Hint:

If the dot product of direction vectors of two lines is zero, the lines are perpendicular.

Answer 1

The Correct Option is D

Step 1: Finding direction vectors of the lines.
From the vector equation, the direction vector of the first line is \[ \vec d_1 = \langle 1, -1, 1 \rangle \] From the symmetric form, the direction vector of the second line is \[ \vec d_2 = \langle 1, 3, 2 \rangle \]
Step 2: Using the dot product formula.
\[ \vec d_1 \cdot \vec d_2 = (1)(1) + (-1)(3) + (1)(2) = 1 - 3 + 2 = 0 \]
Step 3: Interpreting the result.
Since the dot product is zero, the two direction vectors are perpendicular.
Step 4: Conclusion.
Hence, the angle between the two lines is $\dfrac{\pi}{2}$.