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, \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \]

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is \textit{not} the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

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

The Correct Option is A

A line in three-dimensional space cannot be perpendicular to all three axes simultaneously. If a line is perpendicular to all three axes, the direction cosines \( \cos\alpha, \cos\beta, \cos\gamma \) would all be zero, which would violate the fundamental relation of direction cosines: \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. \] The given equation \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \) ensures that at least one of the direction cosines is non-zero, indicating that the line cannot be simultaneously perpendicular to \( x \), \( y \), and \( z \) axes. 
Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).