Question

If \[ O = (0, 0, 0), \quad P = (1, \sqrt{2}, 1), \] then the acute angles made by the line OP with the \( XOY, \, YOZ, \, ZOX \text{ planes are, respectively,} \)

• A. \( 45^\circ, 45^\circ, 60^\circ \)
• B. \( 45^\circ, 60^\circ, 30^\circ \)
• C. \( 60^\circ, 45^\circ, 60^\circ \)
• D. \( 30^\circ, 30^\circ, 45^\circ \)

Hint:

To find the angles between a line and the coordinate planes, use the direction cosines of the line and apply the angle formula with the normal vectors of the planes.

Answer 1

The Correct Option is D

Step 1: Find the direction cosines of the line OP.
The coordinates of the point \( O \) are \( (0, 0, 0) \), and the coordinates of the point \( P \) are \( (1, \sqrt{2}, 1) \). The direction ratios of the line OP are the differences in the coordinates of \( P \) and \( O \), which gives: \[ \text{Direction ratios of OP} = (1, \sqrt{2}, 1). \]
Step 2: Use the direction ratios to find the angles with the planes.
The angles made by the line with the coordinate planes are given by the direction cosines. The direction cosines of the line are given by: \[ \cos \alpha = \frac{1}{\sqrt{1^2 + (\sqrt{2})^2 + 1^2}} = \frac{1}{\sqrt{4}} = \frac{1}{2}, \cos \beta = \frac{\sqrt{2}}{\sqrt{4}} = \frac{\sqrt{2}}{2}, \cos \gamma = \frac{1}{\sqrt{4}} = \frac{1}{2}. \] The acute angles with the planes are the angles between the direction ratios and the normal to the planes. Using the formula for the angle between the line and the plane, we find the angles to be \( 30^\circ, 30^\circ, 45^\circ \). Thus, the correct answer is option (D).