Question

Assertion (A): A line in space cannot be drawn perpendicular to \( x \), \( y \), and \( z \) axes simultaneously. 

Reason (R): For any line making angles \( \alpha, \beta, \gamma \) with the positive directions of \( x \), \( y \), and \( z \) axes respectively, \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \] 

Hint:

In 3D geometry, the direction cosines of a line (\(\cos \alpha, \cos \beta, \cos \gamma\)) satisfy the fundamental relation \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\). This property ensures that no line can be perpendicular to all three axes simultaneously.

Answer 1

A line in three-dimensional space cannot be simultaneously perpendicular to all three axes. Were it to be perpendicular to all three axes, its direction cosines \( \cos\alpha, \cos\beta, \cos\gamma \) would all equal zero, contradicting the fundamental relationship of direction cosines: \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \]

The equation \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \) necessitates that at least one direction cosine be non-zero, which confirms that a line cannot be perpendicular to the \( x \), \( y \), and \( z \) axes concurrently. 

Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).