Question

The angle between the lines \( 2x = 3y - z \) and \( 6x = -y - 4z \) is:

• A. \( 30^\circ \)
• B. \( 45^\circ \)
• C. \( 90^\circ \)
• D. \( 0^\circ \)

Hint:

To find the angle between two lines, use the formula involving the direction ratios and calculate the cosine of the angle.

Answer 1

The Correct Option is C

The general formula for the angle \( \theta \) between two lines with direction ratios \( \langle a_1, b_1, c_1 \rangle \) and \( \langle a_2, b_2, c_2 \rangle \) is given by: \[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \] For the first line \( 2x = 3y - z \), the direction ratios are \( \langle 2, 3, -1 \rangle \). For the second line \( 6x = -y - 4z \), the direction ratios are \( \langle 6, -1, -4 \rangle \). Substitute these values into the formula: \[ \cos \theta = \frac{2 \times 6 + 3 \times (-1) + (-1) \times (-4)}{\sqrt{2^2 + 3^2 + (-1)^2} \sqrt{6^2 + (-1)^2 + (-4)^2}} = 0 \] Thus, the angle between the lines is \( 90^\circ \).