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x-z Plane — Definition, Examples & Formula

x-z Plane

The plane formed by the x-axis and the z-axis.

 

 

See also

x-y plane, y-z plane

Key Formula

y=0y = 0
Where:
  • yy = The coordinate perpendicular to the x-z plane; it equals zero for every point on this plane

Worked Example

Problem: Determine whether the points A(3, 0, 5), B(−2, 0, 7), and C(1, 4, 6) lie on the x-z plane.
Step 1: Recall that a point lies on the x-z plane if and only if its y-coordinate is 0.
y=0y = 0
Step 2: Check point A(3, 0, 5). The y-coordinate is 0, so A lies on the x-z plane.
A=(3,  0,  5)⇒y=0  ✓A = (3,\; 0,\; 5) \quad \Rightarrow \quad y = 0 \; \checkmark
Step 3: Check point B(−2, 0, 7). The y-coordinate is 0, so B also lies on the x-z plane.
B=(−2,  0,  7)⇒y=0  ✓B = (-2,\; 0,\; 7) \quad \Rightarrow \quad y = 0 \; \checkmark
Step 4: Check point C(1, 4, 6). The y-coordinate is 4, which is not zero, so C does not lie on the x-z plane.
C=(1,  4,  6)⇒y=4≠0  ×C = (1,\; 4,\; 6) \quad \Rightarrow \quad y = 4 \neq 0 \; \times
Answer: Points A and B lie on the x-z plane. Point C does not, because its y-coordinate is not zero.

Frequently Asked Questions

What is the equation of the x-z plane?
The equation is simply y = 0. Any point (x, y, z) satisfying this equation — regardless of the values of x and z — lies on the x-z plane.
What is the normal vector to the x-z plane?
The normal vector points in the y-direction. In standard notation it is (0, 1, 0), or equivalently (0, −1, 0). This vector is perpendicular to every line that lies in the x-z plane.

x-z Plane vs. x-y Plane

The x-z plane is defined by y = 0 and contains the x-axis and z-axis. The x-y plane is defined by z = 0 and contains the x-axis and y-axis. Each coordinate plane is named after the two axes it contains, and its equation sets the remaining coordinate to zero. The third standard coordinate plane, the y-z plane, is defined by x = 0.

Why It Matters

The three coordinate planes (x-y, x-z, and y-z) divide three-dimensional space into eight octants — the 3D equivalent of quadrants. In physics and engineering, the x-z plane often represents horizontal ground, with the y-axis pointing upward, which is the convention in many 3D graphics engines and game development frameworks. Understanding coordinate planes is also essential for sketching cross-sections of surfaces and for setting up integrals in multivariable calculus.

Common Mistakes

Mistake: Thinking the x-z plane is defined by x = 0 and z = 0 (setting both named coordinates to zero).
Correction: It is the unnamed coordinate that equals zero. The x-z plane is described by y = 0; x and z can take any values.
Mistake: Confusing the x-z plane with the x-y plane when working in 3D graphics or physics problems that use a "y-up" convention.
Correction: In a y-up coordinate system, the x-z plane is the horizontal (ground) plane (y = 0), while the x-y plane is vertical (z = 0). Always check which axis is designated as "up" in the problem.

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