Question

The sine of the angle between the straight line $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{4-z}{-5}$ and the plane $2x - 2y + z = 5$ is:

• A. $\frac{1}{5\sqrt{2}}$
• B. $\frac{2}{5\sqrt{2}}$
• C. $\frac{3}{50}$\
• D. $\frac{3}\sqrt{50}$

Answer 1

The Correct Option is A

The sine of the angle is given by: \[ \sin \theta = \frac{|A \cdot l + B \cdot m + C \cdot n|}{\sqrt{A^2 + B^2 + C^2} \sqrt{l^2 + m^2 + n^2}}. \] Here, $A = 2, B = -2, C = 1$ (plane), $l = 1, m = 4, n = -5$ (line). Compute: \[ \sin \theta = \frac{|2(1) + (-2)(4) + (1)(-5)|}{\sqrt{2^2 + (-2)^2 + 1^2} \sqrt{1^2 + 4^2 + (-5)^2}} = \frac{1}{5\sqrt{2}}. \]