Math Lab
← All labs

Distance of Origin from a Plane

Three-Dimensional Geometry · Class XII

Plane ax + by + cz = d. Distance from origin is |d| / √(a² + b² + c²).

1
type a value
-55
2
type a value
-55
2
type a value
-55
6
type a value
-1010
Live values
  • Normal magnitude3
  • Distance from O2
  • Foot-of-perp x0.6667
xy
  • Distance from O as d slides

Formulas in this lab

  • Normal magnitude
    a2+b2+c2\sqrt{a^2+b^2+c^2}
  • Distance from O
    ∣d∣a2+b2+c2\dfrac{|d|}{\sqrt{a^2+b^2+c^2}}
  • Foot-of-perp x
    ad/(a2+b2+c2)a d / (a^2+b^2+c^2)
Tip: Distance is a V-shape in d; the plane passes through the origin exactly when d = 0.

Frequently asked questions

▶What is the distance from a point to a plane?

For the plane $ax + by + cz + d = 0$ and the point $(x_0, y_0, z_0)$, the perpendicular distance is $\dfrac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2+b^2+c^2}}$. The absolute value ensures the distance is non-negative.

▶How do I use the point-to-plane distance lab?

Slide the plane coefficients $a, b, c, d$ and the point coordinates. The lab returns the distance live. Try the plane $x + y + z = 3$ and the origin: distance $= 3/\sqrt{3} = \sqrt{3} \approx 1.73$.

▶Why is the absolute value crucial?

Without it, the sign tells you which side of the plane the point lies on — useful for half-space arguments but wrong for a distance, which must be non-negative. JEE often asks for the side as well, separately from the distance. The lab outputs the unsigned distance.

▶Where does this formula appear in real life?

Computer graphics uses it for collision detection: is a player on the right side of a wall? Linear regression's residuals are signed distances from data points to the best-fit plane (hyperplane in higher dimensions). Support Vector Machines maximize the margin computed by this formula.