In three-dimensional geometry, the direction cosines of a vector or a line represent the cosines of the angles that the vector or line makes with the positive x, y, and z-axes, respectively. These cosines are typically denoted by $\text{l}$, $\text{m}$, and $\text{n}$, where:

• $\text{l} = \cos \alpha$ (angle with x-axis)
• $\text{m} = \cos \beta$ (angle with y-axis)
• $\text{n} = \cos \gamma$ (angle with z-axis)

The sum of the squares of the direction cosines of any line is always equal to 1, i.e., $\text{l}^2 + \text{m}^2 + \text{n}^2 = 1$. Let's determine the direction cosines for the z, y, and x-axes.

Direction Cosines for Z-Axis

The z-axis is a line that lies directly along the positive Z direction. Therefore, it makes specific angles with each of the principal axes:

• Angle with x-axis ($\alpha$): The z-axis is perpendicular to the x-axis. So, $\alpha = 90^\circ$.
  • $\text{l} = \cos(90^\circ) = 0$
• Angle with y-axis ($\beta$): The z-axis is perpendicular to the y-axis. So, $\beta = 90^\circ$.
  • $\text{m} = \cos(90^\circ) = 0$
• Angle with z-axis ($\gamma$): The z-axis is aligned with itself. So, $\gamma = 0^\circ$.
  • $\text{n} = \cos(0^\circ) = 1$

Thus, the direction cosines of the z-axis are $(0, 0, 1)$.

Direction Cosines for Y-Axis

The y-axis is a line that lies directly along the positive Y direction. Let's find its angles with the principal axes:

• Angle with x-axis ($\alpha$): The y-axis is perpendicular to the x-axis. So, $\alpha = 90^\circ$.
  • $\text{l} = \cos(90^\circ) = 0$
• Angle with y-axis ($\beta$): The y-axis is aligned with itself. So, $\beta = 0^\circ$.
  • $\text{m} = \cos(0^\circ) = 1$
• Angle with z-axis ($\gamma$): The y-axis is perpendicular to the z-axis. So, $\gamma = 90^\circ$.
  • $\text{n} = \cos(90^\circ) = 0$

Thus, the direction cosines of the y-axis are $(0, 1, 0)$.

Direction Cosines for X-Axis

The x-axis is a line that lies directly along the positive X direction. Here are its angles with the principal axes:

• Angle with x-axis ($\alpha$): The x-axis is aligned with itself. So, $\alpha = 0^\circ$.
  • $\text{l} = \cos(0^\circ) = 1$
• Angle with y-axis ($\beta$): The x-axis is perpendicular to the y-axis. So, $\beta = 90^\circ$.
  • $\text{m} = \cos(90^\circ) = 0$
• Angle with z-axis ($\gamma$): The x-axis is perpendicular to the z-axis. So, $\gamma = 90^\circ$.
  • $\text{n} = \cos(90^\circ) = 0$

Thus, the direction cosines of the x-axis are $(1, 0, 0)$.

Summary of Direction Cosines for Axes

The table below summarizes the direction cosines for the z, y, and x-axes as requested in the question:

Therefore, the sequence of direction cosines for z, y, and x-axis is (0,0,1), (0,1,0), and (1,0,0) respectively.