Question

Find the direction-cosines of the y-axis.

Hint:

This is a standard result worth memorizing. The direction cosines for the coordinate axes are:
- x-axis: (1, 0, 0) since \( \alpha=0^\circ, \beta=90^\circ, \gamma=90^\circ \)
- y-axis: (0, 1, 0) since \( \alpha=90^\circ, \beta=0^\circ, \gamma=90^\circ \)
- z-axis: (0, 0, 1) since \( \alpha=90^\circ, \beta=90^\circ, \gamma=0^\circ \)

Answer 1

Step 1: Understanding the Concept:
Direction cosines of a line or a vector are the cosines of the angles that the line makes with the positive directions of the x, y, and z axes. These angles are typically denoted by \( \alpha, \beta, \) and \( \gamma \), respectively. The direction cosines are \( l = \cos \alpha \), \( m = \cos \beta \), and \( n = \cos \gamma \).
Step 2: Detailed Explanation:
We need to find the direction cosines for the y-axis itself.
Let's determine the angles the y-axis makes with each of the coordinate axes.
1. Angle with the x-axis (\( \alpha \)): The y-axis is perpendicular to the x-axis. Therefore, the angle between them is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.
\[ \alpha = 90^\circ \] 2. Angle with the y-axis (\( \beta \)): The y-axis makes an angle of \( 0^\circ \) with itself.
\[ \beta = 0^\circ \] 3. Angle with the z-axis (\( \gamma \)): The y-axis is perpendicular to the z-axis. Therefore, the angle between them is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.
\[ \gamma = 90^\circ \] Now, we calculate the cosines of these angles to find the direction cosines (\( l, m, n \)).
\[ l = \cos \alpha = \cos(90^\circ) = 0 \] \[ m = \cos \beta = \cos(0^\circ) = 1 \] \[ n = \cos \gamma = \cos(90^\circ) = 0 \] Step 3: Final Answer:
The direction-cosines of the y-axis are (0, 1, 0).