Q.The equation of $z$-axis, are ________.

The $z$-axis is the set of all points where the $x$ and $y$ coordinates are zero, while $z$ can take any value.

Step 1: A point lies on the $z$-axis if and only if its coordinates are of the form $(0, 0, z)$ where $z \in \mathbb{R}$. …

Why these equations represent the $z$-axis

In three-dimensional Cartesian space, every point is uniquely identified by three coordinates $(x, y, z)$. The coordinate axes are the fundamental reference lines:

• The $x$-axis consists of all points where $y = 0$ and $z = 0$ (only the $x$-coordinate varies)
• The $y$-axis consists of all points where $x = 0$ and $z = 0$ (only the $y$-coordinate varies)
• The $z$-axis consists of all points where $x = 0$ and $y = 0$ (only the $z$-coordinate varies)

The $z$-axis is the vertical line passing through the origin, extending infinitely in both directions. Any point on this axis has the form $(0, 0, z)$ where $z$ can be any real number.

Finding the equations

To describe the $z$-axis algebraically, we need to capture the constraint that a point lies on this line.

1. Identify the geometric constraint: A point $(x, y, z)$ lies on the $z$-axis if and only if its $x$-coordinate is zero AND its $y$-coordinate is zero. The $z$-coordinate can take any value.

2. Translate to equations: The condition "$x$-coordinate is zero" translates to the equation $x = 0$. Similarly, "$y$-coordinate is zero" gives $y = 0$.

3. Combine both conditions: Since both conditions must hold simultaneously, the $z$-axis is described by the pair of equations:

   $x = 0, \quad y = 0$ …