Question

Find the direction cosine of Z-axis.

Hint:

It's helpful to memorize the direction cosines of the coordinate axes: - X-axis: (1, 0, 0) - Y-axis: (0, 1, 0) - Z-axis: (0, 0, 1) This saves valuable time in exams.

Answer 1

Step 1: Understanding the Concept:
Direction cosines of a line (or axis) are the cosines of the angles that the line makes with the positive directions of the coordinate axes (X, Y, and Z axes).
If a line makes angles \( \alpha, \beta, \gamma \) with the X, Y, and Z axes respectively, then its direction cosines are \( l = \cos \alpha \), \( m = \cos \beta \), and \( n = \cos \gamma \).
Step 2: Key Formula or Approach:
We need to determine the angles that the Z-axis makes with the X, Y, and Z axes.
Angle with X-axis: \( \alpha \)
Angle with Y-axis: \( \beta \)
Angle with Z-axis: \( \gamma \)
Then, calculate \( \cos \alpha, \cos \beta, \cos \gamma \).
Step 3: Detailed Explanation:
The Z-axis is perpendicular to both the X-axis and the Y-axis.
Therefore, the angle between the Z-axis and the X-axis is \( \alpha = 90^\circ \).
The angle between the Z-axis and the Y-axis is \( \beta = 90^\circ \).
The angle between the Z-axis and itself is \( \gamma = 0^\circ \).
Now, we calculate the cosines of these angles: \[ l = \cos \alpha = \cos(90^\circ) = 0 \] \[ m = \cos \beta = \cos(90^\circ) = 0 \] \[ n = \cos \gamma = \cos(0^\circ) = 1 \] Step 4: Final Answer:
The direction cosines of the Z-axis are (l, m, n), which are (0, 0, 1).